This randomized phase II trial studies how well azacitidine and entinostat before chemotherapy works in treating patients with non-small cell lung cancer that has spread to other places in the body. Drugs used in chemotherapy, such as azacitidine, irinotecan hydrochloride, gemcitabine hydrochloride, docetaxel, and pemetrexed disodium, work in different ways to stop the growth of tumor cells, either by killing the cells, by stopping them from dividing, or by stopping them from spreading. Entinostat may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. Giving azacitidine and entinostat before chemotherapy may work better in treating patients with non-small cell lung cancer.

PRIMARY OBJECTIVES:

I. Percentage of patients progression-free at 6 months from time of randomization.

SECONDARY OBJECTIVES:

I. Progression-free survival (PFS). II. Overall survival (OS).

OUTLINE: Patients are randomized to 1 of 3 treatment arms.

ARM A: Patients receive azacitidine subcutaneously (SC) on days 1-6 and 8-10 and entinostat orally (PO) on days 3 and 10. Treatment repeats every 28 days for 2 courses in the absence of disease progression or unacceptable toxicity. Patients with stable or progressive disease receive chemotherapy of the treating oncologist's choice comprising irinotecan hydrochloride intravenously (IV) on day 1, docetaxel IV on day 1, pemetrexed disodium IV on day 1, or gemcitabine hydrochloride IV on days 1 and 8. Treatment repeats every 21 days in the absence of disease progression or unacceptable toxicity.

ARM B: Patients receive azacitidine PO on days 1-21 and entinostat PO on days 3 and 10. Treatment repeats every 28 days for 2 courses in the absence of disease progression or unacceptable toxicity. Patients with stable or progressive disease receive chemotherapy of the treating oncologist's choice as in Arm A.

ARM C: Patients receive chemotherapy of the treating oncologist's choice as in Arm A.

After completion of treatment, patients are followed up every 3-6 months for 24 months and then yearly thereafter.

Inclusion Criteria:

• Patients must have histologically or cytologically proven non-small cell lung cancer; tumor tissue must be available from all patients prior to initiation of protocol therapy, either from original diagnostic biopsy, or biopsy performed prior to initiation of protocol therapy
• Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as >= 20 mm with conventional techniques or as >= 10 mm with spiral computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam
• Patients must have received 1 prior platinum containing doublet regardless of mutation status
• Patients with targetable mutation i.e. epidermal growth factor receptor (EGFR) or anaplastic lymphoma kinase (ALK), must have been treated with at least 1 prior tyrosine kinase inhibitor (TKI)
• Prior immunotherapy is allowed
• Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 (Karnofsky >= 70%)
• Life expectancy of greater than 12 weeks
• Leukocytes >= 3,000/mcL
• Absolute neutrophil count >= 1,500/mcL
• Platelets >= 100,000/mcL
• Total bilirubin within normal institutional limits
• Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transferase [SGOT])/alanine aminotransferase (ALT) (serum glutamate pyruvate transferase [SGPT]) =< 2.5 X institutional upper limit of normal
• Creatinine within normal institutional limits OR creatinine clearance >= 60 mL/min/1.73 m^2 for patients with creatinine levels above institutional normal
• Women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately
• Ability to understand and the willingness to sign a written informed consent document

Exclusion Criteria:

• Patients who have had chemotherapy or radiotherapy within 4 weeks (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered more than 4 weeks earlier
• Patients who are receiving any other investigational agents
• Patients with uncontrolled brain metastases; patients with brain metastases must have stable neurologic status following local therapy (surgery or radiation) for at least 4 weeks, and must be without neurologic dysfunction that would confound the evaluation of neurologic and other adverse events; patients may be treated with steroids as clinically indicated
• Patients with liver metastases that replace greater than 30% of the liver parenchyma
• History of allergic reactions attributed to compounds of similar chemical or biologic composition to entinostat, azacitidine, mannitol, irinotecan, docetaxel, pemetrexed, or gemcitabine, or other agents used in the study
• Uncontrolled inter-current illness including, but not limited to, ongoing or active infection, New York Heart Association (NYHA) class 3-4 congestive heart failure, unstable angina pectoris, cardiac arrhythmia, or psychiatric illness/social situations that would limit compliance with study requirements
• Pregnant women are excluded from this study; breastfeeding should be discontinued if the mother is treated on this protocol
• Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy are ineligible