Inclusion Criteria:
* The subject has a histologic or cytologic diagnosis of squamous cell carcinoma of the oral cavity, oropharynx, paranasal sinuses, hypopharynx, nasopharynx, or larynx; squamous cell carcinoma of unknown primary in cervical lymph node can be included only if human papillomavirus (HPV) status is positive
* Patients must have refractory, recurrent or metastatic disease, which is deemed to be inoperable
* In case patients received prior systemic therapy within the definitive or metastatic setting, disease progression must be documented following prior therapy; this can be in the recurrent or metastatic setting or in the concurrent setting
* Measurable disease per Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 as determined by the investigator
* A maximum of one prior radiotherapy regimen, curative or palliative, to the head and neck is allowed; if the radiation is combined with chemotherapy, a minimum of 4 months must elapse between the end of radiotherapy and registration; if the radiation is given alone, a minimum of 8 weeks must elapse between the end of radiotherapy and registration; a minimum of 3 weeks must elapse between prior radiation to other areas and registration; treatment areas should be healed with no sequelae from radiation therapy (RT) that would predispose to fistula formation
* The subject has had an assessment of all known disease sites eg, by computerized tomography (CT) scan, magnetic resonance imaging (MRI), bone scan or positron emission tomography (PET)/CT scan as appropriate, within 28 days before the first dose of cabozantinib
* Life expectancy of greater than 3 months
* The subject has an Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1
* Recovery to baseline or ≤ grade 1 Common Terminology Criteria for Adverse Events (CTCAE) version (v.)4.0 from toxicities related to any prior treatments, unless AE(s) are clinically nonsignificant and/or stable on supportive therapy
* Within 7 days before the first dose of cabozantinib: The absolute neutrophil count (ANC) ≥ 1000/mm³ without colony stimulating factor support
* Within 7 days before the first dose of cabozantinib: Platelets ≥ 100,000/mm³
* Within 7 days before the first dose of cabozantinib: Hemoglobin ≥ 9 g/dL
* Within 7 days before the first dose of cabozantinib: Bilirubin ≤ 1.5 x the upper limit of normal (ULN); for subjects with known Gilbert's disease, bilirubin ≤ 3.0 mg/dL
* Within 7 days before the first dose of cabozantinib: Serum albumin ≥ 2.8 g/dl
* Within 7 days before the first dose of cabozantinib: Serum creatinine ≤ 1.5 x ULN or creatinine clearance (CrCl) ≥ 40 mL/min; for creatinine clearance estimation, the Cockcroft and Gault equation should be used
* Within 7 days before the first dose of cabozantinib: Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) ≤ 2.0 x ULN
* Within 7 days before the first dose of cabozantinib: Lipase \< 2.0 x the upper limit of normal and no radiologic or clinical evidence of pancreatitis
* Within 7 days before the first dose of cabozantinib: Urine protein/creatinine ratio (UPCR) ≤ 1
* Within 7 days before the first dose of cabozantinib: Serum phosphorus, calcium, magnesium and potassium ≥ lower limit of normal (LLN)
* The subject is capable of understanding and complying with the protocol requirements and has signed the informed consent document
* Sexually active subjects (men and women) must agree to use medically accepted barrier methods of contraception (eg, male or female condom) during the course of the study and for 4 months after the last dose of study drug(s), even if oral contraceptives are also used; all subjects of reproductive potential must agree to use both a barrier method and a second method of birth control during the course of the study and for 4 months after the last dose of study drug(s)
* Female subjects of childbearing potential must not be pregnant at screening; females of childbearing potential are defined as premenopausal females capable of becoming pregnant (ie, females who have had any evidence of menses in the past 12 months, with the exception of those who had prior hysterectomy); however, women who have been amenorrheic for 12 or more months are still considered to be of childbearing potential if the amenorrhea is possibly due to prior chemotherapy, antiestrogens, low body weight, ovarian suppression or other reasons
Exclusion Criteria:
* Patients who have HPV negative squamous cell carcinoma of unknown primary in cervical lymph node
* The subject has received cytotoxic chemotherapy (including investigational cytotoxic chemotherapy) or biologic agents (eg, cytokines or antibodies) within 4 weeks, or nitrosoureas/mitomycin C within 6 weeks before the first dose of study treatment
* Prior treatment with cabozantinib or pembrolizumab
* Radiation therapy for bone metastasis within 2 weeks, any other external radiation therapy within 4 weeks before the first dose of study treatment; systemic treatment with radionuclides within 6 weeks before the first dose of study treatment; subjects with clinically relevant ongoing complications from prior radiation therapy are not eligible
* Receipt of any type of small molecule kinase inhibitor (including investigational kinase inhibitor) within 14 days before the first dose of study treatment
* The subject has received any other type of investigational agent within 28 days or 5 half-lives, whichever is shorter, before the first dose of study treatment
* 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 the first dose of study treatment; eligible subjects must be neurologically asymptomatic and without corticosteroid treatment at the time of the start of study treatment
* The subject has prothrombin time (PT)/institutional normalized ratio (INR) or partial thromboplastin time (PTT) test ≥ 1.3 x the laboratory ULN within 7 days before the first dose of study treatment
* Concomitant anticoagulation at therapeutic doses with oral anticoagulants (eg, warfarin, direct thrombin and factor Xa inhibitors) or platelet inhibitors (eg, clopidogrel)
* Note: Low-dose aspirin for cardioprotection (per local applicable guidelines), low-dose warfarin (\< 1 mg/day), and low dose, low molecular weight heparins (LMWH) are permitted; anticoagulation with therapeutic doses of LMWH is allowed in subjects who are on a stable dose of LMWH for at least 6 weeks before first dose of study treatment, and who have had no clinically significant hemorrhagic complications from the anticoagulation regimen or the tumor
* The subject has experienced any of the following:
* Clinically-significant GI bleeding within 6 months before the first dose of study treatment
* Hemoptysis of ≥ 0.5 teaspoon (2.5 ml) of red blood within 3 months before the first dose of study treatment
* Any other signs indicative of pulmonary hemorrhage within 3 months before the first dose of study treatment
* The subject has radiographic evidence of cavitating pulmonary lesion(s)
* The subject has tumor invading or encasing any major blood vessels
* The subject has evidence of tumor invading the gastrointestinal (GI) tract (esophagus, stomach, small or large bowel, rectum or anus), or any evidence of endotracheal or endobronchial tumor within 28 days before the first dose of cabozantinib
* The subject has uncontrolled, significant intercurrent or recent illness including, but not limited to, the following conditions:
* Cardiovascular disorders including:
* Congestive heart failure (CHF): New York Heart Association (NYHA) class III (moderate) or class IV (severe) at the time of screening;
* Concurrent uncontrolled hypertension defined as sustained blood pressure (BP) \> 150 mm Hg systolic or \> 100 mm Hg diastolic despite optimal antihypertensive treatment within 7 days of the first dose of study treatment;
* Any history of congenital long QT syndrome;
* Any of the following within 6 months before the first dose of study treatment:
* Unstable angina pectoris;
* Clinically-significant cardiac arrhythmias;
* Stroke (including transient ischemic attack (TIA), or other ischemic event);
* Myocardial infarction;
* GI disorders particularly those associated with a high risk of perforation or fistula formation including:
* Tumors invading the GI tract, active peptic ulcer disease, inflammatory bowel disease (eg, Crohn's disease), diverticulitis, cholecystitis, symptomatic cholangitis or appendicitis, acute pancreatitis or acute obstruction of the pancreatic duct or common bile duct, or gastric outlet obstruction
* Abdominal fistula, GI perforation, bowel obstruction, intra-abdominal abscess within 6 months before randomization
* Note: Complete healing of an intra-abdominal abscess must be confirmed prior to randomization. Also no pre-existing fistula of head and neck area; no pre-existing osteonecrosis of jaw (ONJ)
* Other clinically significant disorders that would preclude safe study participation
* Major surgery within 12 weeks before the first dose of study treatment; complete wound healing from major surgery must have occurred 1 month before the first dose of study treatment; minor surgery (including uncomplicated tooth extractions) within 28 days before the first dose of study treatment with complete wound healing at least 10 days before the first dose of study treatment; subjects with clinically relevant ongoing complications from prior surgery are not eligible
* Corrected QT interval by Fridericia's formula (QTcF) \> 500 msec within 1 month before the first dose of study treatment
* Three electrocardiography (ECG)s must be performed for eligibility determination; if the average of these three consecutive results for QTcF is ≤ 500 msec, the subject meets eligibility in this regard
* Pregnant or lactating females
* Inability to swallow intact tablets or inability to take pembrolizumab or cabozantinib
* Previously identified allergy or hypersensitivity to components of the study treatment formulations
* Patients with a history of other prior malignancy must have been treated with curative intent and must have remained disease-free for 1 year post diagnosis; patients with a prior history of squamous cell or basal carcinoma of the skin or in situ cervical cancer must have been curatively treated
