Cabozantinib S-malate and Nivolumab in Treating Patients With Advanced, Recurrent, or Metastatic Endometrial Cancer

Inclusion Criteria: * Patients must have histologically or cytologically confirmed epithelial endometrial carcinoma; all histologies are accepted; patients with diagnosis of endometrial carcinosarcoma will be enrolled in the exploratory cohort (arm C) and will receive combination of cabozantinib and nivolumab * Patients must have advance, recurrent or metastatic endometrial cancer * Patients must have radiological evidence of disease progression following the most recent treatment * Patients must have measurable disease according Response Evaluation Criteria in Solid Tumors (RECIST) version (v)1.1 criteria * Must have MS/MMR result available at time of registration; MS/MMR status is to be determined per local practice (i.e. immunohistochemistry \[IHC\], polymerase chain reaction \[PCR\], or other methods) * Prior therapy: eligible subjects must have had at least one line of platinum-based chemotherapy; this may be adjuvant therapy or first line of cytotoxic therapy for metastatic disease; prior hormonal therapy for metastatic/recurrent disease, prior targeted therapy, and prior radiotherapy are allowed; no maximum number of previous lines of chemotherapies; concomitant chemo-radiation is not considered as previous line of systemic chemotherapy * Availability of archival tissue for correlative analysis * Age \>=18 years. Because no dosing or adverse event data are currently available on the use of cabozantinib and nivolumab in patients \< 18 years of age, children are excluded from this study, but will be eligible for future pediatric trials * Eastern Cooperative Oncology Group (ECOG) performance status 0-2 * Absolute neutrophil count \>= 1,500/mcL * Platelets \>= 100 x 10\^9/L * Total bilirubin =\< 1.5 ULN (upper limit of normal), unless due to Gilbert's syndrome * Aspartate aminotransferase (AST) (serum glutamic-oxaloacetic transaminase \[SGOT\])/alanine aminotransferase (ALT) (serum glutamate pyruvate transaminase \[SGPT\]) =\< 3 x institutional upper limit of normal * Creatinine =\< 1.5 ULN OR creatinine clearance \>= 50 mL/min/1.73 m\^2 for patients with creatinine levels above institutional normal * Serum albumin \>= 28 g/L * Lipase =\< 2 ULN * Urine protein/ creatinine ratio (UPCR) =\< 1 * Prothrombin time (PT)/ international normalized ratio (INR) and partial thromboplastin time (PTT) test =\< 1.3 ULN * Patient must have disease amenable to biopsy and must agree to have one baseline biopsy * The effects of cabozantinib and nivolumab on the developing human fetus are unknown; women of childbearing potential (WOCBP) must have a negative serum or urine pregnancy test at screening; WOCBP must agree to use adequate contraception (barrier method of birth control or abstinence) prior to study entry and for the duration of study participation; WOCBP should use an adequate method to avoid pregnancy for 7 months after the last dose of investigational drug; women must not be breastfeeding; women of childbearing potential (WOCBP) is defined as any female who has experienced menarche and who has not undergone surgical sterilization (hysterectomy or bilateral oophorectomy) or who is not postmenopausal; menopause is defined clinically as 12 months of amenorrhea in a woman over age 45 in the absence of other biological or physiological causes; in addition, women under the age of 55 must have a documented serum follicle stimulating hormone (FSH) level less than 40 mIU/mL; should a woman become pregnant or suspect she is pregnant while she is participating in this study, she should inform her treating physician immediately * Ability to understand and the willingness to sign a written informed consent document * CROSS-OVER ELIGIBILITY CRITERIA * Patient must provide a tumor biopsy at the time of progression on Arm B; if a patient does not have a tumor lesion amenable to biopsy or it has been unsafe for a biopsy to be performed, cross-over will not be allowed Exclusion Criteria: * Patients who have had chemotherapy (including investigational cytotoxic chemotherapy), biologic agents (e.g. targeted therapy or antibodies) or radiotherapy within 4 weeks prior to the first dose of study treatment * Patients who have not recovered from adverse events attributed to prior anti-cancer therapy (i.e. have residual toxicities \> grade 1, except for alopecia, neuropathy, lymphocytopenia and other non-clinically significant adverse events) * Patients who are receiving any other investigational agents * Patients should be excluded if they have had prior treatment with anti-CTLA-4 antibody or any other antibody or drug specifically targeting T-cell co-stimulation; previous treatment with anti-PD-1, anti-PD-L1 or anti-PD-L2 is allowed and patients will be enrolled in the exploratory cohort (arm C) at the time of progression from last line of treatment (treatment with immune check point inhibitor does not have to necessary be the last line of treatment) * Patients should be excluded if they have had prior treatment with cabozantinib; previous use of other antiangiogenic agents other than cabozantinib is allowed * Any other active malignancy other than the endometrial cancer, that is progressing or requiring active treatment with the exception of basal or squamous cell skin cancer, superficial bladder cancer, carcinoma in situ of any site * Patients with known brain metastases should be excluded from this clinical trial because of their poor prognosis and because of the possible increased risk of bleeding if treatment with antiangiogenic agents is provided; patients with history of brain metastases can enroll provided the brain metastases were removed and controlled with no radiological evidence within the past 6 months * Patients requiring concomitant treatment, in therapeutic doses, with anticoagulants such as warfarin or warfarin-related agents, heparin, thrombin or factor Xa inhibitors, antiplatelet agents (e.g. clopidogrel) or new oral anticoagulants; low-dose aspirin (=\< 81 mg/day), and prophylactic low molecular weight heparin (LMWH) are permitted * History of allergic reactions attributed to compounds of similar chemical or biologic composition to cabozantinib or nivolumab * Patients require chronic concomitant treatment of strong CYP450 3A4 inducers (e.g. dexamethasone, phenytoin, carbamazepine, rifampicin, rifabutin, rifapentine, phenobarbital, St. John's wort) or inhibitors (e.g. ketoconazole, miconazole, itraconazole, voriconazole, posaconazole, clarithromycin, telithromycin, indinavir, saquinavir, ritonavir, nelfinavir, amprenavir, fosamprenavir nefazodone, lopinavir, troleandomycin, mibefradil and conivaptan); because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated list, medical reference texts such as the Physicians' Desk Reference may also provide this information; as part of the enrollment/informed consent procedures, the patient will be counseled on the risk of interactions with other agents, and what to do if new medications need to be prescribed or if the patient is considering a new over-the-counter medicine or herbal product * The subject has experienced any of the following: * Clinically-significant gastrointestinal 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 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 XL184 (cabozantinib) * Subject with extensive pelvic mass at risk of fistulization, or history of bowel obstruction within 3 months prior to the proposed first dose of study treatment * 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) \> 140 mmHg systolic, or \> 90 mmHg 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 * Thromboembolic event requiring therapeutic anticoagulation (Note: subjects with a venous filter \[e.g. vena cava filter\] are not eligible for this study) * Gastrointestinal disorders particularly those associated with a high risk of perforation or fistula formation including: * Any of the following within 28 days before the first dose of study treatment * Intra-abdominal tumor/metastases invading GI mucosa * Active peptic ulcer disease, * Inflammatory bowel disease (including ulcerative colitis and Crohn's disease), diverticulitis, cholecystitis, symptomatic cholangitis or appendicitis * Malabsorption syndrome * Any of the following within 6 months before the first dose of study treatment: * Abdominal fistula * Gastrointestinal perforation * Intra-abdominal abscess; Note: complete resolution of an intra-abdominal abscess must be confirmed prior to initiating treatment with cabozantinib even if the abscess occurred more than 6 months before the first dose of study treatment * Other disorders associated with a high risk of fistula formation including percutaneous endoscopic gastrostomy (PEG) tube placement * Other clinically significant disorders such as: * Active infection requiring systemic treatment within 28 days before the first dose of study treatment * Serious non-healing wound/ulcer/bone fracture within 28 days before the first dose of study treatment * History of organ transplant * Concurrent uncompensated hypothyroidism or thyroid dysfunction within 7 days before the first dose of study treatment * History of major surgery as follows: * Major surgery within 3 months of the first dose of cabozantinib if there were no wound healing complications or within 6 months of the first dose of cabozantinib if there were wound complications * Minor surgery within 1 month of the first dose of cabozantinib if there were no wound healing complications or within 3 months of the first dose of cabozantinib if there were wound complications * In addition, complete wound healing from prior surgery must be confirmed at least 28 days before the first dose of cabozantinib irrespective of the time from surgery * Known active human immunodeficiency virus (HIV), acquired immunodeficiency syndrome (AIDS) related illness, or hepatitis B or C infection * Administration of a live vaccine within 4 weeks prior to start of protocol therapy * Subjects with diagnosis of immunodeficiency or who are receiving systemic steroid therapy or any other form of immunosuppressive therapy within 7 days prior to the first dose of trial treatment; the following are exceptions to this exclusion criteria: intranasal, inhaled, topical steroids, or local steroids injections (e.g. intra-articular injection); systemic corticosteroids at physiologic dose not to exceed 10 mg/day of prednisone or equivalent; steroids as premedication for hypersensitivity reactions (e.g., CT scan premedication) * History of autoimmune disease, such as, but not restricted to: rheumatoid arthritis, inflammatory bowel disease, systemic lupus erythematous, ankylosing spondylitis, scleroderma, or multiple sclerosis requiring treatment within the last two years; patients with vitiligo or diabetes are not excluded; replacement therapy (e.g. thyroxine, insulin, or physiologic corticosteroid replacement therapy for adrenal or pituitary insufficiency, etc.) is not considered a form of systemic treatment; patients with recent history of thyroiditis; subjects with remote history (greater than 5 years) of thyroiditis are not excluded * Psychiatric illness/social situations that would limit compliance with study requirements * The subject has a corrected QT interval calculated by the Fridericia formula (QTcF) \> 500 ms within 28 days before randomization; Note: if initial QTcF is found to be \> 500 ms, two additional electrocardiogram (EKG)s separated by at least 3 minutes should be performed; if the average of these three consecutive results for QTcF is =\< 500 ms, the subject meets eligibility in this regard * Patient is not able to swallow pills * Pregnant women are excluded from this study because XL184 and nivolumab are agents with the potential for teratogenic or abortifacient effects; because there is an unknown but potential risk for adverse events in nursing infants secondary to treatment of the mother with XL184 and nivolumab, breastfeeding should be discontinued if the mother is treated with XL184 and nivolumab