The main purpose of this study is to see whether the combination of two drugs called pembrolizumab and vorinostat can help people with advanced lung cancer. Researchers also want to find out if the combination of pembrolizumab and vorinostat is safe and tolerable. This study will compare the effects of the combination of two drugs called pembrolizumab and vorinostat with the effects of pembrolizumab alone. The U.S. Food and Drug Administration (FDA) has approved pembrolizumab for use to treat a deadly skin cancer called melanoma and lung cancer and vorinostat to treat some forms of blood and lymph node cancers.

A Phase I/Randomized Phase II clinical trial of pembrolizumab and vorinostat in Eastern Cooperative Oncology Group (ECOG) 0-1 patients with immune therapy naïve and immune therapy pretreated locally advanced or metastatic NSCLC who have progressed through one prior line of therapy.

The begins with a phase I dose escalation utilizing the modified continuous reassessment method (O'Quigley, Pepe, & Fisher, 1990). This would be followed by a phase I expansion at the maximum tolerated dose (MTD) in 18 NSCLC patients who have been previously treated with anti-PD-1 or anti-PD-L1 therapy. In parallel, a separate phase II arm will randomize 70 patients to a pembrolizumab alone group and a pembrolizumab plus vorinostat group.

Inclusion Criteria:

• Willing and able to provide written informed consent/assent for the trial.
• Be ≥ 18 years of age on day of signing informed consent.
• Have measurable disease based on Response Evaluation Criteria in Solid Tumors (RECIST) 1.1.
• Have archival tissue where available. Those participants enrolled on the phase 1 escalation trial where archival tissue is not available will undergo a fresh biopsy where clinically feasible after discussion with the sponsor.
• In addition, participants enrolled on the phase 1 dose escalation, phase 1 expansion or Phase II trial must be willing and able to provide tissue from a newly obtained core or excisional biopsy of a tumor lesion.
• Randomized Phase II: Tumor proportional score of PD-L1 ≥1%.
• Have a performance status of 0 or 1 on the Eastern Cooperative Oncology Group (ECOG) Performance Scale.
• Demonstrate adequate organ function.
• Have a histologic or cytologic diagnosis of Stage IV non-small cell lung cancer (NSCLC).
• Phase I/IB (Pre-treated): Have progression from at least one prior line of therapy. Maintenance therapy following platinum doublet-based chemotherapy is not considered as a separate regimen of therapy. Participants who received platinum-containing adjuvant, neoadjuvant or definitive chemoradiation therapy given for locally advanced disease, and developed recurrent (local or metastatic) disease within 6 months of completing therapy are eligible for these arms. Participants with recurrent disease ≥ 6 months after completing a platinum-containing adjuvant, neoadjuvant or definitive chemoradiation therapy given for locally advanced disease, who also subsequently progressed during or after a systemic regimen given to treat the recurrence, must have received another treatment in the first-line metastatic setting.
• Randomized Phase II: Be treatment naïve in the stage IV setting, with the exception of participants whose tumors harbor an activation mutation (including but not limited to EGFR, ALK, ROS1) and were previously treated with targeted therapy. Participants who received platinum-containing adjuvant, neoadjuvant or definitive chemoradiation therapy given for locally advanced disease, and developed recurrent (local or metastatic) disease < 6 months of completing therapy are ineligible for this arm. Subjects with recurrent disease ≥ 6 months after completing a platinum-containing adjuvant, neoadjuvant or definitive chemoradiation therapy given for locally advanced disease, who also subsequently progressed during or after a systemic regimen given to treat the recurrence, are eligible for this arm.
• Females of childbearing potential should be willing to use 2 methods of birth control or be surgically sterile, or abstain from heterosexual activity for the course of the study through 120 days after the last dose of study medication.
• Males should agree to use an adequate method of barrier contraception starting with the first dose of study therapy through 120 days after the last dose of study therapy.

Exclusion Criteria:

• Is currently participating in and receiving study therapy or has participated in a study of an investigational agent and received study therapy or used an investigational device within 4 weeks of the first dose of treatment.
• Has a diagnosis of immunodeficiency or is receiving systemic steroid therapy at doses ≥ 10 mg prednisone or any other form of systemic immunosuppressive therapy within 7 days prior to the first dose of trial treatment. Participants are permitted to use topical, ocular, intra-articular, intranasal, and inhalational corticosteroids (with minimal systemic absorption). Physiologic replacement doses of systemic corticosteroids are permitted, even if >= 10 mg/day prednisone equivalents. A brief course (≤28 days) of corticosteroids for prophylaxis (e.g., contrast dye allergy) or for treatment of non-autoimmune conditions (e.g., delayed-type hypersensitivity reaction caused by contact allergen) is permitted.
• Has a known history of tuberculosis (TB) Disease (Mycobacterium tuberculosis).
• Hypersensitivity to pembrolizumab, vorinostat or any of its excipients.
• Participants enrolled on the phase II randomized trial, who have had prior treatment with a PD1 or PDL1 inhibitor, anti-CTLA 4 antibody or any other antibody or drug that specifically targets immune checkpoint pathway (i.e. not "immune therapy naïve"). -Note: For those enrolled in the phase I dose escalation, prior use of a PD1 or PDL1, anti-CTLA4 antibody or any other antibody or drug that specifically targets immune checkpoint pathway is allowed. For those enrolled in the phase IB, prior use of a PD1 or PDL1, anti-CTLA4 antibody or any other antibody or drug that specifically targets immune checkpoint pathway is required. For all participants in all phases, prior use of a vaccine for treatment of cancer is allowed.
• Participants enrolled in the phase Ib expansion who have never previously been treated with a PD1 or PDL1 inhibitor, anti-CTLA 4 antibody or any other antibody or drug that specifically targets immune checkpoint pathway in the past (i.e. not "pre-treated").
• Participants who have received thoracic radiation >30Gy within 6 months of the first dose of pembrolizumab.
• Patients taking any HDACi other than vorinostat.
• Has had a prior anti-cancer monoclonal antibody (mAb) within 4 weeks prior to study Day 1 or who has not recovered (i.e., ≤ Grade 1 or at baseline) from adverse events due to agents administered more than 4 weeks earlier.
• Has had prior chemotherapy with 3 weeks, or targeted small molecule therapy or radiation therapy within 2 weeks prior to study Day 1 or who has not recovered (i.e., ≤ Grade 1 or at baseline) from adverse events due to a previously administered agent. Note: Patients with ≤ Grade 2 neuropathy are an exception to this criterion and may qualify for the study. Note: Patients with any grade alopecia are an exception to this criterion and may qualify for the study.
• If participant received major surgery, they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy.
• Has a known additional malignancy that is progressing or requires active treatment. Exceptions include basal cell carcinoma of the skin or squamous cell carcinoma of the skin that has undergone potentially curative therapy, or in situ cervical cancer.
• Has known active central nervous system (CNS) metastases and/or carcinomatous meningitis. Previously treated brain metastases may be an exception if stable and specific other criteria are met.
• Has active autoimmune disease that has required systemic treatment in the past 2 years. Replacement therapy is not considered a form of systemic treatment. Patients are permitted to enroll if they have vitiligo, type I diabetes mellitus, residual hypothyroidism due to autoimmune condition only requiring hormone replacement, psoriasis not requiring systemic treatment, or conditions not expected to recur in the absence of an external trigger.
• Has an active infection requiring systemic therapy.
• Is pregnant or breastfeeding, or expecting to conceive or father children within the projected duration of the trial, starting with the pre-screening or screening visit through 120 days after the last dose of trial treatment.
• Has a known history of Human Immunodeficiency Virus (HIV) (HIV 1/2 antibodies).
• Has known active Hepatitis B (e.g., HBsAg reactive) or Hepatitis C (e.g., HCV RNA [qualitative] is detected).
• Has received a live vaccine within 30 days of planned start of study therapy. Note: Seasonal influenza vaccines for injection are generally inactivated flu vaccines and are allowed; however intranasal influenza vaccines (e.g., Flu-Mist®) are live attenuated vaccines, and are not allowed.
• Has a history of, or any evidence of active non-infectious pneumonitis that required or requires steroids.
• Has evidence of interstitial lung disease
• Has a history of symptomatic (NYHA class II-IV) heart failure

All adverse events are recorded from C1D1 and for up to 2 years