Abexinostat in Combination With Pembrolizumab in Patients With Advanced Solid Tumor Malignancies

Inclusion Criteria: * Patient \>= 18 years of age at the time of study enrollment. * Has histologically confirmed locally advanced or metastatic solid tumor malignancy with one of the following tumor types: * Urothelial carcinoma * Melanoma * Non-small cell lung cancer * Small cell lung cancer * Non-pulmonary squamous cell carcinoma * Head and neck squamous cell carcinoma * MSI-high solid tumor, with MSI-high status defined by microsatellite instability testing by Polymerase chain reaction (PCR), loss of mismatch repair proteins by Immunohistochemistry (IHC), or Clinical Laboratory Improvement Amendments (CLIA)-approved next generation sequencing test. * Gastric and gastro-esophageal junction adenocarcinoma * Merkel cell carcinoma * Thymic carcinoma * Mesothelioma * Measurable disease by RECIST 1.1 criteria. * Dose Expansion only: * Disease progression during or within 3 months of last dose of most recent line of prior anti-PD-1/PD-L1-based treatment with a pattern of progression as defined as one of the following: * Primary Resistance (Cohort A): Progressive disease as best response to prior treatment with anti-PD-1/PD-L1 treatment * Acquired Resistance (Cohort B): Stable disease, partial response or complete response to prior anti-PD-1/PD-L1 treatment with subsequent disease progression during or within 3 months following last dose of anti-PD-1/PD-L1 treatment * A tumor biopsy is mandatory unless * There is no safely accessible lesion, OR * Archival metastatic tumor tissue is available with sufficient quantity that was obtained following the last dose of most recent systemic therapy. * Has Eastern Cooperative Oncology Group (ECOG) performance status of 0-2. * Has adequate baseline organ function, as demonstrated by the following: * Serum creatinine =\< 1.5 x institutional upper limit of normal (ULN) or calculated creatinine clearance \> 50 mL/min * Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN) * Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 2.5 x institutional ULN (patients with hepatic metastases must have AST/ALT =\< 5 x ULN). * Absolute neutrophil count (ANC) \>= 1.5 x 10\^9/L. * Hemoglobin \>= 8 g/dL. * Platelet count \>= 75 x 10\^9/L. * Agrees to use acceptable contraceptive methods for the duration of time on the study, and continue to use acceptable contraceptive methods for 6 months (after the last treatment with study treatment. * Has provided signed informed consent before initiation of any study-specific procedures or treatment. * Men treated or enrolled on this protocol must agree to use adequate contraception the duration of study participation and 3 months after completion of study drug administration. Exclusion Criteria: * Has persistent clinically significant toxicities (grade \>= 2; per National Cancer Institute Common Terminology Criteria for Adverse Events (NCI CTCAE) version 5) from previous anticancer therapy (excluding alopecia which is permitted and excluding grades 2 and 3 laboratory abnormalities if they are not associated with symptoms, are not considered clinically significant by the Investigator, and can be managed with available medical therapies). * Has a history of (non-infectious) pneumonitis/interstitial lung disease that required steroids or current pneumonitis/ interstitial lung disease * Has a diagnosis of clinically significant immunodeficiency. * Has received external-beam radiation or another systemic anticancer therapy within 14 days or 5 half-lives (whichever is shorter) before start of study treatment. * Has received treatment with an investigational drug or monoclonal antibody within 28 days prior to study treatment administration. For classes of investigational agents that are not known to have prolonged toxicities, the washout time may be decreased to 14 days at the discretion of the principal investigator. * Has received previous treatment with a histone deacetylase (HDAC) inhibitor. * Has an additional active malignancy that may confound the assessment of the study endpoints. Patients with the following concomitant neoplastic diagnoses are eligible: nonmelanoma skin cancer and carcinoma in situ including transitional cell carcinoma, anal carcinoma, and melanoma in situ). * Has clinically significant cardiovascular disease including, but not limited to: * Uncontrolled or any New York Heart Association class 3 or 4 congestive heart failure * Uncontrolled angina, history of myocardial infarction, unstable angina or stroke within 6 months before study entry * Clinically significant arrhythmias not controlled by medication. * Has a history of untreated brain, or leptomeningeal, metastases (central nervous system (CNS) imaging is not required before study entry unless there is a clinical suspicion of CNS involvement). Subjects with previously treated brain metastases may participate provided: * They are stable (without evidence of progression by imaging for at least four weeks and any neurologic symptoms have returned to baseline) * They have no evidence of new or enlarging brain metastases (confirmed by imaging within 28 days of the first dose of study drug) * They are not using steroids for at least 7 days before the first dose of study drug. This exception does not include leptomeningeal metastases, which is excluded regardless of clinical stability. * Has a condition requiring systemic treatment with either corticosteroids (\> 10 mg daily prednisone equivalents) or other immunosuppressive medications within 14 days before study treatment. * Has uncontrolled intercurrent illness including, but not limited to: * Uncontrolled infection * Disseminated intravascular coagulation * Psychiatric illness/social situations that would limit compliance with study requirements. * Has known history positive status for human immunodeficiency virus or chronic active hepatitis B or hepatitis C (screening not required). * Has any medical condition which in the opinion of the Investigator places the patient at an unacceptably high risk for toxicities. * Use of prohibited medication within 7 days or 5 half-lives, whichever is shorter, prior to first dose of study drug