Androgen Deprivation Therapy (ADT) and Pembrolizumab for Advanced Stage Androgen Receptor-positive Salivary Gland Carcinoma

Inclusion Criteria: * Written informed consent and HIPAA authorization for release of personal health information. NOTE: HIPAA authorization may be included in the informed consent or obtained separately. * Age ≥ 18 years at the time of consent. * Locally advanced, recurrent, or metastatic salivary gland carcinoma that is not amenable to curative surgery or radiation * ECOG Performance Status of 0 or 1 within 28 days prior to registration. * Local, pathologic testing of androgen receptor-positive salivary gland carcinoma will be performed as standard of care. Archival tissue must be available for central confirmation of androgen receptor-positive disease and for correlative studies. AR positivity will be defined according to IHC staining of tumor tissue with at least 20% of tumor staining positive with moderate intensity (1+ or greater). * Measurable disease according to RECIST v1.1 for solid tumors within 28 days prior to registration. * For patients who have been treated with prior therapy, patients must have documented progression of disease on their prior therapy for entry into the study. * Patients with prior chemotherapy, radiation, or surgery as part of curative intent therapy are allowed. Any number of prior lines of systemic therapy is permitted for entry into this study so long as prior therapy did not include anti-androgen therapy or immune checkpoint blockade. * If prior cancer treatment, the subject must have recovered from toxic effects of prior cancer treatment (other than alopecia) to ≤ Grade 1. * Adequate organ function as defined below; all screening labs to be obtained within 28 days prior to registration. * Absolute neutrophil count (ANC) ≥1500/µL * Platelets ≥100,000/µL * Hemoglobin ≥9.0 g/dL or ≥5.6 mmol/L * Creatinine (Cr) OR Measured or calculated creatinine clearance (GFR can also be used in place of Cr or creatinine clearance) ≤1.5 × ULN OR ≥30 mL/min for participant with creatinine levels \>1.5 × institutional ULN * Total bilirubin ≤1.5 ×ULN OR direct bilirubin ≤ULN for participants with total bilirubin levels \>1.5 × ULN * AST (SGOT) and ALT (SGPT) ≤2.5 × ULN (≤5 × ULN for participants with liver metastases) o International normalized ratio (INR) OR prothrombin time (PT) \& aPTT ≤1.5 × ULN unless participant is receiving anticoagulant therapy as long as PT or aPTT is within therapeutic range of intended use of anticoagulants * A male participant must agree to use contraception during the treatment period and for at least 8 months after the last dose of study treatment and refrain from donating sperm during this period. * Females of childbearing potential must have a negative serum pregnancy test within 72 hours prior to registration. NOTE: Females are considered of child bearing potential unless they are surgically sterile (have undergone a hysterectomy, bilateral tubal ligation, or bilateral oophorectomy) or they are naturally postmenopausal for at least 12 consecutive months * Females of childbearing potential and males with partners of childbearing potential must be willing to abstain from heterosexual activity or to use a highly effect form of contraception from the time of informed consent until 8 months after treatment discontinuation. * As determined by the enrolling physician or protocol designee, ability of the subject to understand and comply with study procedures for the entire length of the study Exclusion Criteria: * Women of childbearing age with a positive serum pregnancy test within 72 hours prior to study registration. * Has received prior therapy with an anti-PD-1, anti-PD-L1, or anti-PD-L2 agent or with an agent directed to another stimulatory or co-inhibitory T-cell receptor (eg, CTLA-4, OX40, CD137). * Has received prior androgen deprivation therapy including orchiectomy, gonadotropin-releasing hormone (GnRH) agonists/antagonists, androgen receptor blocker, abiraterone, or enzalutamide. * Has received prior systemic anti-cancer therapy including investigational agents within 14 days prior to registration. * Has had an allogenic tissue or solid organ transplant. * Has received prior palliative radiotherapy within 7 days of start of study treatment. Participants must have recovered from all radiation-related toxicities and require less than 10mg of prednisone (or equivalent corticosteroid) daily. * Has received a live vaccine or live-attenuated vaccine within 28 days prior to the first dose of study drug. Administration of killed vaccines is allowed. * Has a diagnosis of immunodeficiency or is receiving chronic systemic steroid therapy (in dosing exceeding 10 mg daily of prednisone equivalent) or any other form of immunosuppressive therapy within 7 days prior to the first dose of study drug. * Has a known additional malignancy that is progressing or has required active treatment within the past 2 years. Note: Participants with basal cell carcinoma of the skin, squamous cell carcinoma of the skin, or carcinoma in situ (e.g. breast ductal carcinoma in situ, cervical cancer in situ) that have undergone potentially curative therapy are not excluded. * Has known active CNS metastases and/or carcinomatous meningitis. Participants with previously treated brain metastases may participate provided they are radiologically stable, i.e. without evidence of progression for at least 14 days by repeat imaging (note that the repeat imaging should be performed during study screening), clinically stable, and without requirement of steroid treatment for at least 14 days prior to first dose of study treatment. * Has ≥Grade 3 hypersensitivity to pembrolizumab and/or any of its excipients. * 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, levothyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment. * Has a history of (non-infectious) pneumonitis/interstitial lung disease that required steroids or has current pneumonitis/interstitial lung disease. * Has an active infection requiring systemic therapy. * Has a known history of Human Immunodeficiency Virus (HIV). * Has a known history of active TB (Bacillus Tuberculosis). * Has a history or current evidence of any condition, therapy, or laboratory abnormality that might confound the results of the study, interfere with the subject's participation for the full duration of the study, or is not in the best interest of the subject to participate, in the opinion of the treating investigator. * Has known psychiatric or substance abuse disorders that would interfere with cooperation with the requirements of the trial. * Is pregnant or breastfeeding or expecting to conceive or father children within the projected duration of the study, starting with the screening visit through 120 days after the last dose of trial treatment.