Inclusion Criteria:

Patients with NSCLC:

1. Has histologically- or pathologically-confirmed recurrent/metastatic NSCLC.
2. If has adenocarcinoma, required to have previously been tested for anaplastic lymphoma kinase (ALK) rearrangements and epidermal growth factor receptor (EGFR) mutations, with results available for collection in this study, and, if positive, has been treated with prior EGFR or ALK therapy.
3. Received at least 1 chemotherapeutic regimen in the advanced/metastatic setting and experienced documented, unequivocal progressive disease by either RECIST 1.1 or clinical assessment.
4. Patients with NSCLC enrolled in Cohort 1 of the Expansion Phase should not have been previously treated with a PD-1/PD-L1-blocking antibody Patients in Expansion Phase, Cohorts 2 and 3
5. Previously treated with a PD-1/PD-L1-blocking antibody and experienced experienced documented, unequivocal radiographic progression of disease by irRECIST, or similar criteria during or within 12 weeks after last dose of such treatment. Patients must have received at least 6 weeks of PD-1/PD-L1 therapy for Cohort 3. Patients with Melanoma:
6. In addition to having been previously treated with a PD-1/PD-L1-blocking antibody (inclusion #5), has a histologically- or cytologically-confirmed diagnosis of unresectable or metastatic melanoma and experienced unequivocal progressive disease during treatment with a BRAF inhibitor if BRAF V600 mutation-positive. Treatment with BRAF inhibitor may occur AFTER treatment with the checkpoint inhibitor. Patients in Expansion Phase, Cohort 4 (Colorectal Cancer)
7. Received at least 1 chemotherapeutic regimen in the advanced/metastatic setting and experienced documented, unequivocal progressive disease by either RECIST 1.1 or clinical assessment. Must have documented mismatch repair-proficient colon cancer as determined by either immunohistochemistry for mismatch repair proteins or PCR-based functional microsatellite instability. Patients with colorectal cancer enrolled in Cohort 4 should not have been previously treated with a PD-1/PD-L1-blocking antibody (i.e., pembrolizumab, nivolumab, MEDI36MEDI4376, or GNE PDL1 [MPDL3280A]) All Patients
8. Aged 18 years or older on the day written informed consent is given.
9. If has brain metastases, must have stable neurologic status following local therapy for at least 4 weeks without the use of steroids or on stable or decreasing dose of ≤10 mg daily prednisone (or equivalent), and must be without neurologic dysfunction that would confound the evaluation of neurologic and other AEs.
10. Evidence of locally recurrent or metastatic disease based on imaging studies within 28 days before the first study drug dose:
    • At least 1 measurable lesion ≥20 mm by conventional techniques or ≥10 mm by spiral CT scan or MRI, with the last imaging performed within 28 days before the first study drug dose. If there is only 1 measurable lesion and it is located in previously irradiated field, it must have demonstrated unequivocal progression according to RECIST, version 1.1.
11. If receiving radiation therapy, has a 2-week washout period following completion of the treatment prior to receiving the first study drug dose and continues to have at least 1 measureable lesion, per above criterion.
12. ECOG performance status of 0 or 1.
13. Has acceptable, applicable laboratory parameters.
14. Female subjects must not be pregnant.
15. If male, agrees to use an adequate method of contraception

Experienced resolution of toxic effect(s) of the most recent prior chemotherapy to Grade 1 or less (except alopecia). If patient underwent major surgery or radiation therapy of >30 Gy, they must have recovered from the toxicity and/or complications from the intervention. 17. Willing to have fresh tumor samples collected during screening and at other time points designated as mandatory, per the Schedule of Study Assessments. 18. Able to understand and give written informed consent and comply with study procedures.

Exclusion Criteria:

Patients meeting any of the following criteria are not eligible for study participation:

1. Diagnosis of immunodeficiency or receiving systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to the first dose of study drug. The use of physiologic doses of corticosteroids may be approved after consultation with the Sponsor.
2. Active autoimmune disease that has required systemic treatment in past 2 years (i.e., with disease modifying agents, corticosteroids, or immunosuppressive drugs). Replacement therapy (e.g., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency) is not considered a form of systemic treatment.
3. History of interstitial lung disease (ILD).
4. Allergy to benzamide or inactive components of entinostat.
5. History of allergies to any active or inactive ingredients of pembrolizumab or severe hypersensitivity (>= Grade 3) to pembroluzumab.
6. History or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the study, interfere with the patient's participation for the full duration of the study, or is not in the best interest of the patient to participate, in the opinion of the treating Investigator, including, but not limited to:
   • Myocardial infarction or arterial thromboembolic events within 6 months prior to baseline or severe or unstable angina, New York Heart Association (NYHA) Class III or IV disease, or a QTc interval > 470 msec.
   • Uncontrolled heart failure or hypertension, uncontrolled diabetes mellitus, or uncontrolled systemic infection.
   • Another known additional malignancy that is progressing or requires active treatment (excluding adequately treated basal cell carcinoma, squamous cell of the skin, cervical intraepithelial neoplasia [CIN]/cervical carcinoma in situ or melanoma in situ, or ductal carinoma in situ of the breast). Prior history of other cancer is allowed, as long as there is no active disease within the prior 5 years.
   • Has a history of (non-infectious) pneumonitis that required steroids or current pneumonitis.
   • Active infection requiring systemic therapy.
   • Known active central nervous system (CNS) metastases and/or carcinomatous meningitis.

   Note: Patients with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging [using the identical imaging modality for each assessment, either MRI or CT scan] for at least 4 weeks prior to the first dose of study drug and any neurologic symptoms have returned to baseline), have no evidence of new or enlarging brain metastases, and are not using steroids for at least 2 weeks prior to the first dose of study drug or are on stable or decreasing dose of ≤10 mg daily prednisone (or equivalent). This exception does not include carcinomatous meningitis which is excluded regardless of clinical stability.

7. Known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the study.
8. Currently participating 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.
9. Received a live virus vaccination within 30 days of the first dose of treatment.
10. Prior anti-cancer monoclonal antibody (mAb) within 4 weeks prior to baseline or who has not recovered (i.e., ≤Grade 1 or at baseline) from AEs due to agents administered more than 4 weeks earlier.
11. Prior chemotherapy, targeted small molecule therapy, or radiation therapy within 2 weeks prior to study baseline or who has not recovered (i.e., ≤Grade 1 or at baseline) from AEs due to a previously administered agent.

    Note: Patients with ≤Grade 2 neuropathy or ≤Grade 2 alopecia are an exception to this criterion and may qualify for the study.

    Note: If patient underwent major surgery, they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy.

12. Received transfusion of blood products (including platelets or red blood cells) or administration of colony stimulating factors (including granulocyte-colony stimulating factor [G-CSF], granulocyte macrophage-colony stimulating factor [GM-CSF], or recombinant erythropoietin) within 4 weeks prior to the first dose of study drug.
13. Currently receiving treatment with any other agent listed on the prohibited medication list such as valproic acid, or other systemic cancer agents within 14 days of the first dose of treatment.
14. If female, is pregnant, breastfeeding, or expecting to conceive, or if male, expect to father children within the projected duration of the study, starting with the screening visit through 120 days after the last dose of study drug.
15. Known history of human immunodeficiency virus (HIV) (HIV 1/2 antibodies).
16. Known active hepatitis B (e.g., hepatitis B surface antigen-reactive) or hepatitis C (e.g., hepatitis C virus ribonucleic acid [qualitative]).
17. For CRC expansion cohort, no prior history of malignant bowel obstruction requiring hospitalization in the 6 months prior to enrollment
18. For the CRC expansion cohort, no history of uncontrolled ascites, defined as symptomatic ascites and/or repeated paracenteses for symptom control in the past 3 months