Axitinib and Pembrolizumab in Subjects With Advanced Alveolar Soft Part Sarcoma and Other Soft Tissue Sarcomas

Inclusion Criteria: 1. Patients must have histologically confirmed sarcoma with pathology review required for any outside samples. 2. The following histologies may be enrolled without prior treatment: * alveolar soft part sarcoma, * clear cell sarcoma, * epithelioid hemangioendothelioma, and * chordoma. 3. The following histologies may be enrolled only if refractory to anthracycline-based chemotherapy or if the patient refuses to undergo standard of care treatment: * synovial sarcoma, * rhabdomyosarcoma, * malignant peripheral nerve sheath tumors, * dedifferentiated, pleomorphic or myxoid/round cell liposarcoma, * leiomyosarcoma, * malignant phylloides tumor, * high grade undifferentiated pleomorphic sarcomas (HGUPS/MFH), * angiosarcoma, * spindle cell sarcoma, not otherwise specified (NOS) * malignant myoepithelioma. 4. The following histologies may be enrolled only if refractory to at least one line of chemotherapy or if the patient refuses to undergo standard of care treatment: * solitary fibrous tumor/hemangiopericytoma. 5. The following histologies may be enrolled only if refractory to at least first-line targeted therapy or if the patient refuses to undergo standard of care treatment: * gastrointestinal stromal tumors, * extraskeletal myxoid chondrosarcoma, * PEComa. 6. Primary tumors of bone including Ewing's sarcoma, osteosarcoma, and dedifferentiated chondrosarcoma may only be enrolled if there are measurable target lesions occurring in soft tissue and they are refractory to standard of care anthracycline-based chemotherapy. 7. Any other histology or standard of care therapy not specifically addressed will be reviewed by the principal investigator and pathologist for final determination of eligibility. 8. Measurable disease as defined by RECIST v1.1. 9. Radiographic progression as defined by RECIST v1.1, based on comparison between two radiographic studies no greater than 6 months apart. 10. Inability to undergo complete resection of the disease by surgery. 11. Adequate organ function as defined: * Hematological * Absolute neutrophil count (ANC) ≥1,000 / microliter (mcL) * Platelets ≥75,000 / mcL * Hemoglobin ≥8 g/dL without transfusion or erythropoietin (EPO) dependency (within 7 days of assessment) * Renal * Serum creatinine ≤1.5 X upper limit of normal (ULN) OR Measured or calculated creatinine clearance ≥ 60 mL/min for subject with creatinine levels \> 1.5 X institutional ULN. (GFR can also be used in place of creatinine or CrCl). Creatinine clearance should be calculated per institutional standard. * Hepatic * Serum total bilirubin ≤ 1.5 X ULN OR Direct bilirubin ≤ ULN for subjects with total bilirubin levels \> 1.5 ULN. * Aspartate Aminotransferase (AST/SGOT) and Alanine Transaminase (ALT/SGPT) ≤ 2.5 X ULN OR ≤ 5 X ULN for subjects with liver metastases. * Albumin \>2.5 mg/dL * Coagulation * International Normalized Ratio (INR) or Prothrombin Time (PT) ≤1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants. * Activated Partial Thromboplastin Time (aPTT) ≤1.5 X ULN unless subject is receiving anticoagulant therapy as long as PT or PTT is within therapeutic range of intended use of anticoagulants. 12. Age ≥ 16 years. 13. Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1. 14. Patients must consent and be willing to undergo three core needle biopsies at baseline, prior to starting Cycle 3, and at off-study. At least one tumor site must be amenable to biopsy in the judgment of the interventional radiologist. 15. Female subject of childbearing potential should have a negative urine or serum pregnancy within 72 hours prior to receiving the first dose of study medication. If the urine test is positive or cannot be confirmed as negative, a serum pregnancy test will be required. 16. Females of child bearing potential that are sexually active must agree to either practice 2 medically accepted highly effective methods of contraception at the same time or abstain from heterosexual intercourse from the time of signing the informed consent through 120 days after the last dose of study drug. See Appendix G for protocol-approved highly effective methods of contraceptive combinations. Subjects of childbearing potential are those who have not been surgically sterilized or have not been free from menses for \> 1 year. * Negative test for pregnancy is required of females of child-bearing potential; A female of child bearing potential is any woman, regardless of sexual orientation or whether they have undergone tubal ligation, who meets the following criteria: 1. has not undergone a hysterectomy or bilateral oophorectomy; or 2. has not been naturally postmenopausal for at least 24 consecutive months (i.e., has had menses at any time in the preceding 24 consecutive months or 730 days). * Conception while on treatment must be avoided 17. 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 therapy. Prior history of vasectomy does NOT replace requirement for contraceptive use. 18. Suitable venous access to allow for all study related blood sampling 19. Ability to understand and willingness to sign a written informed consent document. 20. For minors that are 16 to 18 years of age, assent and parental (or legally acceptable representative) written informed consent must be obtained. Exclusion Criteria: 1. Prior therapy with axitinib. Patients are permitted to have received prior tyrosine kinase inhibitor (TKI) therapy including imatinib, sunitinib, pazopanib, or similar. Patients may have received prior Programmed death 1 (PD-1)/Programmed death-ligand 1 (PD-L1) directed therapy. 2. Hypersensitivity to axitinib, pembrolizumab or any of its excipients. 3. Patients may not be receiving any other investigational agents (within 4 weeks prior to Cycle 1, day 1). 4. Prior anti-cancer monoclonal antibody (mAb) within 4 weeks prior to Cycle 1, day 1 or has not recovered (i.e., ≤ Grade 1 or at baseline) from adverse events due to agents administered more than 4 weeks earlier. 5. Patient has had prior chemotherapy, targeted small molecule therapy, or radiation therapy within 2 weeks prior to Cycle 1, Day 1 or has not recovered (i.e., ≤ Grade 1 or at baseline) from adverse events due to agents administered more than 4 weeks earlier. Subjects with ≤ Grade 2 neuropathy are an exception to this criterion and may qualify for the study. Note: If subject received major surgery, they must have recovered adequately from the toxicity and/or complications from the intervention prior to starting therapy. 6. Additional known 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. 7. Patients with end-organ dysfunction as defined in inclusion criterion (i.e. #11 above). 8. Patients with bone-only lesions. 9. Patients with underlying immune deficiency, chronic infections including HIV, hepatitis, or tuberculosis (TB) or autoimmune disease. 10. Patients with underlying hematologic issues including bleeding diathesis, known previous GI bleeding requiring intervention within the past 6 months, active pulmonary emboli or deep vein thromboses (DVT) that are not stable on anticoagulation regimen. 11. Has known history of, or any evidence of active, non-infectious pneumonitis. 12. Has known active central nervous system (CNS) metastases and/or carcinomatous meningitis or leptomeningeal disease. Subjects with previously treated brain metastases may participate provided they are stable (without evidence of progression by imaging for at least four weeks prior to the first dose of trial treatment 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 7 days prior to trial treatment. This exception does not include carcinomatous meningitis which is excluded regardless of clinical stability. 13. Concomitant (or receipt of) treatment with medications that may affect the metabolism of pembrolizumab and/or axitinib within 7 days prior to Cycle 1, day 1 of axitinib. 14. 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. 15. 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. 16. Any uncontrolled, intercurrent illness including but not limited to ongoing or active infection, symptomatic congestive heart failure, unstable angina pectoris, cardiac arrhythmia 17. Prolonged corrected QT (QTc) interval on Screening EKG \>475 ms. 18. Ejection Fraction \<40% by 2D echocardiogram (ECHO) at Screening. 19. Any serious medical or psychiatric illness/condition including substance use disorders likely in the judgment of the Investigator(s) to interfere or limit compliance with study requirements/treatment. 20. Has active autoimmune disease that has required systemic treatment in the past 2 years (i.e. with use of disease modifying agents, corticosteroids or immunosuppressive drugs). Replacement therapy (eg., thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment.