This phase II trial studies the best dose and side effects of sorafenib tosylate and nivolumab in treating patients with liver cancer that cannot be removed by surgery (unresectable), has spread to nearby tissues or lymph nodes (locally advanced) or to other places in the body (metastatic). Sorafenib tosylate may stop the growth of cancer cells by blocking some of the enzymes needed for cell growth. Immunotherapy with monoclonal antibodies, such as nivolumab, may help the body's immune system attack the cancer, and may interfere with the ability of tumor cells to grow and spread. Giving sorafenib tosylate and nivolumab may work better in treating patients with liver cancer.

PRIMARY OBJECTIVES:

I. Maximum tolerated dose (MTD) of sorafenib tosylate (sorafenib) in combination with standard dose nivolumab with Child Pugh A-B7 liver function. (Part 1: Escalation Cohort).

II. Safety in participants with Child-Pugh B liver function. (Part 2: Child-Pugh B Escalation Cohort)

SECONDARY OBJECTIVES:

I. Safety and tolerability of combination overall. (Parts 1 and 2).

II. Rate of immune-related adverse event (irAE) for combination overall and in participants with Child-Pugh B liver function. (Parts 1 and 2).

III. Overall response rate (ORR) by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 in participants with Child-Pugh B liver function. (Parts 1 and 2).

IV. Duration of response (DOR), progression free survival (PFS), and overall survival (OS) for escalation cohort and Child-Pugh B expansion cohort and overall. (Parts 1 and 2).

EXPLORATORY OBJECTIVES:

I. Relationship between peripheral blood mononuclear cell (PBMC) immune cell subset frequencies, baseline liver function, and clinical outcomes.

II. Relationship between PBMC T cell receptor (TCR) clonotype frequency and diversity, baseline liver function, and clinical outcomes.

III. Tumor tissue immune cell subsets and TCR clonotype frequency and diversity in pre-treatment archival tumor tissue samples.

IV. Tumor and immune cell programmed death-ligand 1 (PD-L1) status in pre-treatment archival tumor tissue samples and relationship to clinical outcomes.

V. Changes in hepatitis B virus (HBV) and/or hepatitis C virus (HCV) viral load on treatment.

VI. Alpha-fetoprotein (AFP) changes on treatment and relationship to clinical outcomes.

VII. Relationship between clinical outcomes and clinicopathologic features including race/ethnicity, etiology of liver disease including HBV/HCV status, baseline liver function, presence of cirrhosis, macrovessel invasion, extrahepatic spread, site(s) of metastatic disease, prior treatment history including prior radiation and arterial therapies.

OUTLINE: This is a dose-escalation and expansion study of sorafenib tosylate.

DOSE-ESCALATION (Part 1):

Between 3-12 participants will be enrolled in Part 1. Participants receive sorafenib tosylate orally (PO) once daily (QD) or twice daily (BID) on days 1-28, and nivolumab intravenously (IV) over 30 minutes on days 1 and 15. Cycles repeat every 28 days in the absence of disease progression or unacceptable toxicity.

EXPANSION COHORT (Part 2).

Participants receive nivolumab IV over 30 minutes on days 1 and 15, and sorafenib tosylate once daily (QD) or twice daily (BID) on days 1-28. Cycles repeat every 28 days in the absence of disease progression or unacceptable toxicity.

After completion of study treatment, participants are followed up at 30 and 100 days, then every 3 months for up to 2 years after the last dose.

Inclusion Criteria:

1. Histologic or cytologic diagnosis of unresectable, locally advanced and/or metastatic hepatocellular carcinoma (HCC) not amenable to curative surgery, transplantation, or ablative therapies based upon assessment of treating investigator.

   a. For patients without prior histologic or cytologic diagnosis, radiographic diagnosis is allowed provided patients meet American Association for the Study of Liver Diseases (AASLD) criteria for radiographic diagnosis.

2. Radiographically measurable disease by Response Evaluation Criteria in Solid Tumors (RECIST) version 1.1 in at least one site not previously treated with chemoembolization, radioembolization, radiation, or other local/liver-directed procedures (i.e. must have at least one measurable target lesion, either within the liver or in a measurable metastatic site); a new area of tumor progression within or adjacent to a previously-treated lesion, if clearly measurable by a radiologist, is acceptable.

3. Untreated/pretreatment archival tumor tissue must be available for correlative analyses

4. Age at least 18 years at enrollment.

5. Eastern Cooperative Oncology Group (ECOG) 0 or 1 at enrollment

6. At least 4 weeks after any prior chemoembolization, radioembolization, local ablative therapies, or hepatic radiation and recovery to =< grade 1 treatment-related toxicity.

7. At least 6 weeks after any major surgery including prior hepatic resection and recovery to =< grade 1 treatment-related toxicity.

8. At least 7 days after minor surgery (such as central venous access) or biopsy and recovery to =< grade 1 treatment-related toxicity

9. At least 2 weeks after any prior palliative radiation (e.g. to focal metastatic lesion such as bone metastases) and recovery to =< grade 1 treatment-related toxicity.

10. Blood pressure =< 140/90 mm Hg with or without anti-hypertensive therapy

    a. Patients may be rescreened after initial ineligibility if due to elevated blood pressure, if adequately medically managed within approximately 30 days.

11. Adequate baseline organ and marrow function as defined below:

    1. Adequate bone marrow function:

       • Absolute neutrophil count at least 1,200/microliter (mcL).
       • Platelets at least 50,000/mcL.
       • Hemoglobin at least 9 g/dL.

    2. Adequate hepatic function:

       • Part 1: Total bilirubin less than 2.6 mg/dL or 2 times upper limit of normal (ULN), whichever is higher, and albumin at least 2.5 g/dL, if otherwise meets criteria for Child Pugh A or B7.
       • Part 2: Total bilirubin less than 3.9 mg/dL or 3 times ULN, whichever is higher, and albumin at least 2.0 g/dL, if otherwise meets criteria for Child Pugh B.
       • Both Parts: Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase (SGOT)) and alanine aminotransferase (ALT) (serum glutamic-pyruvic transaminase (SGPT)) less than 5 X ULN, International normalized ratio (INR) less than 1.7.

    3. Creatinine less than 1. 5 X ULN and/or creatinine clearance >= 60 mL/min.

12. Child-Pugh A or B7 (Part 1); Child-Pugh B7-9 (Part 2).

13. If HBV surface antigen (sAg) and/or core antibody (Ab) positive, must be treated with appropriate antiviral therapy according to institutional practice with HBV deoxyribonucleic acid (DNA) by polymerase chain reaction (PCR) less than 500 IU/mL.

14. If clinical or histologic diagnosis of cirrhosis and/or clinical or radiographic evidence esophageal or gastric varices, must have had esophagogastroduodenoscopy (EGD) surveillance and adequate endoscopic therapy according to institutional standards.

15. Able to swallow and retain oral medications

16. Women of child-bearing potential (WOCBP) must have a negative pregnancy test within 28 days before study enrollment.

17. WOCBP and male partners of WOCBP must agree to use two methods of contraception until at least 5 months after last dose of each study drug for WOCBP subjects, and 7 months for male partners of WOCBP.

18. Able to understand and willingness to provide informed consent, and the willingness to comply with the requirements of the protocol.

    1. Subjects must have signed and dated an Institutional Review Board (IRB)/Independent Ethics Committees (IEC) approved written informed consent form in accordance with regulatory and institutional guidelines and before the performance of any protocol related procedures that are not part of standard of care.

Exclusion Criteria:

1. Any prior systemic therapy for HCC.

2. Known fibrolamellar or mixed HCC-cholangiocarcinoma histology.

3. Requirement for paracentesis to control ascites within 6 months before enrollment.

   a. Ascites which is not clinically detectable or mild on stable doses of diuretics during screening is allowed provided meets criteria for Child Pugh A or B7 (Part 1) or B7-9 (Part 2).

4. Symptomatic hepatic encephalopathy requiring medication (such as lactulose or rifaximin) (Part 1) or any hospitalization for encephalopathy within 6 months before enrollment (Part 1 or 2).

   1. Hepatic encephalopathy that is adequately controlled on stable doses of lactulose and/or rifaximin per assessment of treating investigator is allowed in Part 2, provided no hospitalization for encephalopathy within 6 months before enrollment.
   2. Medications such as lactulose used for other indications (e.g. constipation) are allowed in both Part 1 and 2.

5. History of upper gastrointestinal (GI) bleeding from esophageal and/or gastric varices within 12 months before enrollment.

6. Requirement for systemic corticosteroids unless used for adrenal replacement, acute therapy for asthma or bronchitis exacerbation (=< 2 weeks), or premedication for contrast allergy.

   a. Topical, intranasal, or inhaled steroids are not excluded.

7. Active autoimmune condition requiring systemic immunosuppressive medication.

8. Known human immunodeficiency virus (HIV) infection.

9. Active coinfection with HBV plus hepatitis delta (D) virus (HDV) or HCV:

   1. Both hepatitis B and C as evidenced by detectable HBV surface antigen or HBV DNA and detectable HCV ribonucleic acid (RNA).
   2. Hepatitis D infection (HDV antibody positive) in subjects with detectable hepatitis B surface antigen or HBV DNA.

10. Prior allogeneic transplant of any solid organ or bone marrow/stem cells.

11. Symptomatic hypothyroidism without replacement.

    a. Patients may be rescreened after initiating adequate replacement therapy

12. History of seizure disorder requiring antiepileptic medication or brain metastases with seizures.

13. Non-healing wound, ulcer, non-healing traumatic bone fracture, or abscess within 30 days of enrollment.

    a. Nondisplaced, uncomplicated pathologic fracture due to tumor may be eligible provided adequately treated with radiation, surgery or other treatments with full recovery based upon investigator assessment.

14. Central or necrotic lung metastases.

15. Known brain or leptomeningeal metastases.

16. Uncontrolled hypertension (systolic pressure > 140 mm Hg and/or diastolic pressure > 90 mm Hg (National Cancer Institute (NCI)-Common Terminology Criteria for Adverse Events (CTCAE) v4.0) on repeated measurement) despite optimal medical management.

17. Active or clinically significant cardiac disease including:

    1. Congestive heart failure - New York Heart Association (NYHA) > class II.
    2. Active coronary artery disease including unstable or newly diagnosed angina or myocardial infarction within 6 months prior to study entry.
    3. Cardiac arrhythmias requiring anti-arrhythmic therapy other than beta blockers or digoxin.
    4. Corrected QT interval (QTc) (Fridericia) > 450 msec on two consecutive electrocardiograms (ECGs) (baseline ECG should be repeated if QTc is found to be > 450 msec).

18. Subject with any pulmonary hemorrhage/bleeding event of NCI-CTCAE v4.0 grade 2 or higher within 6 months before first dose of study treatment any other hemorrhage/bleeding event of NCI-CTCAE v4.0 grade 3 or higher within 6 months before first dose of study treatment.

19. Subjects with arterial or venous thrombotic or embolic, such as cerebrovascular accident (including transient ischemic attacks), myocardial infarction, or deep venous thrombosis (DVT) within 6 months of informed consent.

    1. Tumor or bland thrombus in hepatic vasculature is not an exclusion provided hepatic function criteria are met.
    2. Asymptomatic thromboembolic events such as incidentally-detected sub-segmental pulmonary emboli or superficial thromboses are not an exclusion provided the patient does not require treatment with therapeutic anticoagulation.

20. Subjects who have used strong CYP3A4 inducers (eg, phenytoin, carbamazepine, phenobarbital, St. John's wort (hypericum perforatum), dexamethasone at a dose of greater than 16 mg daily, or rifampin [rifampicin], and/or rifabutin) within 28 days before first dose of study treatment.

21. Subjects who require therapeutic anticoagulation or anti-platelet therapy.

    1. Low dose aspirin (=< 100 mg/day) is allowed
    2. Prophylactic doses of low molecular weight heparin (LMWH) are allowed if approved by study chair or designee.

22. Subjects with any previously untreated and concurrent cancer that is distinct in primary site or histology from HCC except cervical cancer in-situ, treated nonmelanoma skin cancers, localized prostate cancer not requiring systemic therapy undergoing surveillance, or superficial bladder tumor; subjects surviving a cancer that was curatively treated and without evidence of disease for more than 2 years before enrollment are allowed provided that cancer therapy was completed at least 2 years prior to study entry (date of the informed consent form).

23. Any uncontrolled intercurrent illness including, but not limited to: ongoing or active infection requiring antibiotic therapy, pulmonary disease impairing functional status or requiring oxygen, impairment in gastrointestinal function that may affect or alter absorption of oral medications (such as malabsorption or history of gastrectomy or bowel resection), or uncontrolled diarrhea.

24. Known or suspected allergy or hypersensitivity to any of the study drugs, study drug classes, or excipients of the formulations given during the course of this trial

25. Women who are pregnant or breast-feeding at enrollment

26. Inability to comply with the protocol and/or not willing or not available for follow-up assessments.

27. Any condition which, in the investigator's opinion, makes the subject unsuitable for trial participation