Apatinib With Pembrolizumab in Previously Treated Advanced Malignancies

Inclusion Criteria: * Male or female subject aged ≥ 18 years. * One of the following advanced solid malignancies which qualifies for standard of care pembrolizumab treatment per FDA approval: * Locally advanced or metastatic urothelial carcinoma that has progressed during or following platinum-based chemotherapy or within 12 months of neoadjuvant/adjuvant platinum-based therapy. Patients may have received any amount of platinum-based therapy. * Unresectable or metastatic MSI-H or dMMR solid tumors that have progressed during or following prior treatment and have no satisfactory alternative treatment options (including MSI-H or dMMR colorectal cancer that has progressed following treatment with a fluoropyrimidine, oxaliplatin, and irinotecan). * Recurrent locally advanced or metastatic, gastric or gastroesophageal junction (GEJ) adenocarcinoma expressing PD-L1 (as determined by an FDA-approved test) that have progressed on or after two or more systemic therapies, including fluoropyrimidine- and platinum-containing chemotherapy and, if appropriate, HER2/neu-targeted therapy. * Patients must have available and be willing to provide formalin fixed paraffin embedded tissue sample from archival tissue (patients who can't provide archival tissue will be offered an optional biopsy; lack of tissue will not be exclusionary). * Have measurable disease based on RECIST 1.1. (For Phase 2 Subjects Only) * Female subject of childbearing potential should have a negative urine or serum pregnancy test within 72 hours prior to receiving the first dose of study drug. If the urine test is positive, a serum pregnancy test will be required. * Female subjects 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. Subjects of childbearing potential are those who have not been surgically sterilized or have not been free from menses for at least 1 year. * Male subjects should agree to use an adequate method of contraception starting with the first dose of study therapy through 120 days after the last dose of study drug. * ECOG Performance Status ≤ 1. * Adequate organ function as defined in the protocol * Recovery to baseline or Grade ≤ 1 CTCAE v4 from toxicities related to any prior treatments, unless AE(s) are clinically nonsignificant and/or stable on supportive therapy. * Patients must be able to provide informed consent and be willing to sign an approved consent form that conforms to federal and institutional guidelines Exclusion Criteria: * Previous treatment with and disease progression on a combination of a VEGF inhibitor and an immune checkpoint inhibitor. Patients who have been treated with and have progressed on a single agent VEGF inhibitor OR an immune checkpoint inhibitor will not be excluded. * Current use of immunosuppressive medication, EXCEPT for the following: * Topical, ocular, intra-articular, intranasal, and inhaled corticosteroids. * Systemic corticosteroids at physiologic doses ≤ 10 mg/day of prednisone equivalent. * Steroids as premedication for hypersensitivity reactions. * Active autoimmune disease that might deteriorate when receiving an immune-stimulatory agent per treating physician's clinical judgment. Subjects with type 1 diabetes, vitiligo, psoriasis, hypothyroidism, or hyperthyroidism not requiring immunosuppressive medications are eligible. * Prior organ transplant including allogenic hematopoietic stem cell transplant. * Active infection requiring intravenous antibiotics (must be completed prior to registration). * Known brain metastases or cranial epidural disease unless adequately treated with radiotherapy and/or surgery (including radiosurgery) and stable for at least 4 weeks before first dose of study treatment. Eligible subjects must be neurologically asymptomatic and without corticosteroid treatment at the time of first dose of study treatment. * Uncontrolled, significant intercurrent or recent illness including, but not limited to, the following conditions: * Cardiovascular disorders: * Uncontrolled hypertension defined as sustained blood pressure (BP) \> 140 mm Hg systolic or \> 90 mm Hg diastolic despite optimal antihypertensive treatment. * Congestive heart failure New York Heart Association Class 3 or 4, unstable angina pectoris, serious cardiac arrhythmias. * Stroke (including transient ischemic attack \[TIA\]), myocardial infarction (MI), or other ischemic event, or thromboembolic event (eg, deep venous thrombosis, pulmonary embolism) within 6 months before first dose. * Any history of congenital long QT syndrome. * Presence of a non-healing wound. * Other clinically significant disorders that would preclude safe study participation including colitis, inflammatory bowel disease, pneumonitis, pulmonary fibrosis, psychiatric conditions with active suicidal ideation within the past year; or laboratory abnormalities that may increase the risk associated with study participation or study treatment administration and, in the judgment of the investigator, would make the patient inappropriate for entry into this study. * History of allergic reactions attributed to compounds of similar chemical or biologic composition to apatinib or pembrolizumab. * History of severe hypersensitivity reaction to any monoclonal antibody. * Known history of testing positive for HIV or known acquired immunodeficiency syndrome. * Known history of hepatitis B virus (HBV) or hepatitis C virus (HCV) infection at screening (positive HBV surface antigen or detectable HCV RNA if anti-HCV antibody screening test positive). * Live vaccine within 4 weeks of the first dose of pembrolizumab and while on trial is prohibited. * Packed red blood cell transfusions or erythropoietin therapy within 14 days prior to the study enrollment unless erythropoietin therapy has been used to maintain a stable condition for at least 1 month prior to the enrollment. * Palliative radiotherapy within 2 weeks of the first dose of study treatment. * Major surgery within 4 weeks of first dose of study medications. Minor procedures (e.g. port placement, endoscopy with intervention) within 2 weeks of first dose of study medications are allowed. * Chemotherapy, targeted small molecule therapy, or investigational therapy within 4 weeks of the first dose of study treatment. * Known prior severe hypersensitivity to investigational product or any component in its formulations, including known severe hypersensitivity reactions to monoclonal antibodies (NCI CTCAE v4 Grade ≥ 3). * Subjects taking prohibited medications as described in the protocol with the exception of systemic corticosteroids as defined in the protocol. A washout period of at least 5 elimination half-lives (or as clinically indicated) should occur for prohibited medications prior to the start of treatment.