Cabozantinib-S-Malate in Treating Patients With Hormone-Resistant Metastatic Prostate Cancer

Inclusion Criteria: * The subject has histologically confirmed prostate adenocarcinoma with radiologic evidence of metastases * If patient are on anti-androgens, these should be discontinued, at least 4 weeks prior for flutamide and at least 6 weeks for bicalutamide or nilutamide * At least 14 days should have elapsed from prior radiation therapy to bone metastases from prostate cancer * The patient has received a maximum of one prior chemotherapy regimen for metastatic prostate cancer * Patients must demonstrate disease progression on or after most recent systemic therapy, either by prostate-specific antigen (PSA), new bone metastases or by measurable disease criteria per Response Evaluation Criteria in Solid Tumors (RECIST) guidelines * Patients should have received either luteinizing hormone-releasing hormone (LHRH) analogue, or LHRH analogue and anti- androgen for metastatic prostate cancer * The patient has an Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * Bisphosphonate therapy can be continued if started prior to protocol enrollment * Patients must have blood pressure (BP) readings \< 150/90 prior to enrollment * Absolute neutrophil count (ANC) \>= 1500/mm\^3 without colony stimulating factor support * Platelets \>= 100,000/mm\^3 * Hemoglobin \>= 9 g/dL * Bilirubin =\< 1.5 x the upper limit of normal (ULN); for subjects with known Gilbert's disease, bilirubin =\< 3.0 mg/dL * Serum albumin \>= 2.8 g/dl * Serum creatinine =\< 1.5 x ULN or creatinine clearance (CrCl) \>= 50 mL/min; for creatinine clearance estimation, the Cockcroft and Gault equation should be used * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 2.5 x ULN if no liver involvement, or =\< 5 x ULN with liver involvement * Lipase \< 1.5 x the upper limit of normal (except for subjects with adenocarcinoma of the pancreas) * Urine protein/creatinine ratio (UPCR) =\< 1 * Serum phosphorus \>= lower limit of normal (LLN) * The subject is capable of understanding and complying with the protocol requirements and has signed the informed consent document * Patients participating in this trial must also be eligible and willing to sign consent for participation in \[2'-18F\]-1-(2'-deoxy-2'-fluoro-beta-D-arabinofuranosyl)thymine (FMAU) and 18F-flouride positron emission tomography (PET) scans done under separate protocols * Sexually active subjects (men) must agree to use medically accepted barrier methods of contraception (eg, male condom, or diaphragm with spermicidal gel) 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 by the female partner; all subjects of reproductive potential must agree to use both a barrier method and a second method of birth control * Projected life expectancy of at least 6 months * No prior history of other malignancies in the last 3 years, except for squamous and basal cell skin cancer Exclusion Criteria: * The subject has received cytotoxic chemotherapy (including investigational cytotoxic chemotherapy) or biologic agents (eg, cytokines or antibodies) within 3 weeks, or nitrosoureas/mitomycin C within 6 weeks before the first dose of study treatment * Prior treatment with cabozantinib * The subject has received radiation therapy: * To the thoracic cavity or gastrointestinal tract within 3 months of the first dose of study treatment * To bone or brain metastasis within 14 days of the first dose of study treatment * To any other site(s) within 28 days of the first dose of study treatment * The subject has received radionuclide treatment within 6 weeks of the first dose of study treatment * The subject has received prior treatment with a small molecule kinase inhibitor or a hormonal therapy (including investigational kinase inhibitors or hormones) within 14 days or five half-lives of the compound or active metabolites, whichever is longer, before the first dose of study treatment; patients receiving LHRH or gonadotropin-releasing hormone (GnRH) agonists to maintain castrate levels of testosterone or patients on bisphosphonate/denosumab, may be maintained on these agents * The subject has received any other type of investigational agent within 28 days before the first dose of study treatment * The subject has not recovered to baseline or Common Terminology Criteria for Adverse Events (CTCAE) =\< grade 1 from toxicity due to all prior therapies except alopecia and other non-clinically significant adverse events (AEs) * The subject has a primary brain tumor * The subject has active brain metastases or epidural disease (Note: Subjects with brain metastases previously treated with whole brain radiation or radiosurgery or subjects with epidural disease previously treated with radiation or surgery who are asymptomatic and do not require steroid treatment for at least 2 weeks before starting study treatment are eligible; neurosurgical resection of brain metastases or brain biopsy is permitted if completed at least 3 months before starting study treatment; baseline brain scans are not required to confirm eligibility) * The subject has prothrombin time (PT)/international normalized ratio (INR) or partial thromboplastin time (PTT) test results at screening \>= 1.3 x the laboratory ULN * The subject requires concomitant treatment, in therapeutic doses, with anticoagulants such as warfarin or warfarin-related agents, heparin, thrombin or Factor xabans (Xa) inhibitors, or antiplatelet agents (eg, clopidogrel); low dose aspirin (=\< 81 mg/day), low-dose warfarin (=\< 1 mg/day), and prophylactic low molecular weight heparin (LMWH) are permitted * The subject has experienced any of the following within 3 months before the first dose of study treatment: * Clinically-significant hematemesis or gastrointestinal bleeding * Hemoptysis of \>= 0.5 teaspoon (2.5 ml) of red blood * Any other signs indicative of pulmonary hemorrhage * The subject has radiographic evidence of cavitating pulmonary lesion(s) * The subject has tumor in contact with, invading or encasing major blood vessels * 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 BP \> 140 mmHg systolic, or \> 90 mmHg diastolic despite optimal antihypertensive treatment (BP must be controlled at screening) * 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 at the time of screening * Intra-abdominal tumor/metastases invading gastrointestinal (GI) mucosa * Active peptic ulcer disease * Inflammatory bowel disease (including ulcerative colitis and Crohn's disease), diverticulitis, cholecystitis, symptomatic cholangitis or appendicitis * Any of the following within 6 months before the first dose of study treatment: * History of abdominal fistula * Gastrointestinal perforation * Bowel obstruction or gastric outlet obstruction * Intra-abdominal abscess. Note: Complete resolution of an intraabdominal abscess must be confirmed prior to initiating treatment with cabozantinib even if the abscess occurred more that 6 months ago * GI surgery (particularly when associated with delayed or incomplete healing) within 28 days; Note: Complete healing following abdominal surgery must be confirmed prior to initiating treatment with cabozantinib even if surgery occurred more that 28 days ago * Other disorders associated with a high risk of fistula formation including percutaneous endoscopic gastrostomy (PEG) tube placement within 3 months before the first dose of study therapy or concurrent evidence of intraluminal tumor involving the trachea and esophagus * Other clinically significant disorders such as: * Active infection requiring systemic treatment * Serious non-healing wound/ulcer/bone fracture * History of organ transplant * Concurrent uncompensated hypothyroidism or thyroid dysfunction * History of major surgery within 4 weeks or minor surgical procedures within 1 week before randomization * The subject is unable to swallow tablets * The subject has a corrected QT interval calculated by the Fridericia formula (QTcF) \> 470 ms within 28 days before randomization * The subject has a previously identified allergy or hypersensitivity to components of the study treatment formulation * The subject is unable or unwilling to abide by the study protocol or cooperate fully with the investigator or designee * The subject has had evidence within 2 years of the start of study treatment of another malignancy, which required systemic treatment