Cabozantinib-S-Malate in Treating Patients With Refractory Thyroid Cancer

Inclusion Criteria: * Patients must have histologically or cytologically confirmed papillary thyroid cancer, follicular thyroid cancer or hurthle cell thyroid cancer (cancer, follicular variant of papillary thyroid cancers or any of the above mixed histology will be allowed; these patients will be enrolled on Arm A of the trial); patients with the following aggressive histologies will be enrolled on Arm B of the trial; tall cell, insular or poorly differentiated thyroid cancer * Patients must have measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as \>= 2.0 cm with conventional techniques or as \>= 1.0 cm (or \>= 1.5 cm in short axis for lymph node) with spiral computed tomography (CT) scan, magnetic resonance imaging (MRI), or calipers by clinical exam * Patients must have radioactive iodine (RAI)-refractory/resistant disease as defined by one or more of the following criteria: * One or more measurable lesions that do not demonstrate RAI uptake * One or more measurable lesions progressive by Response Evaluation Criteria in Solid Tumors (RECIST) 1.1 within 12 months of prior RAI therapy * One or more measurable lesion present after cumulative RAI dose of \>= 600 mCi * Patients must have "progressed on" up to 2 lines of prior vascular endothelial growth factor receptor (VEGFR)-targeted therapy (including but not limited to sorafenib, sunitinib, vandetanib, pazopanib, or lenvatinib) as defined by progressive disease per RECIST while receiving VEGFR-targeted therapy; Note: patients who progressed on kinase inhibitor that target VEGFR and MET will not be eligible * Prior external beam radiation, systemic cytotoxic chemotherapy or BRAF- or non-VEGFR-targeted therapies will be allowed, provided that \>= 4 weeks has elapsed since receiving prior treatment and they have recovered to =\< grade 1 treatment-related toxicities * Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * Absolute neutrophil count \>= 1,500/mcL * Platelet count \>= 100,000/mcL * Hemoglobin \>= 9 g/dL * Thyroid stimulating hormone (TSH) \< lower limit of normal (LLN) (patients not able to tolerate TSH suppression can be granted an exception) * Total bilirubin =\< 1.5 x upper limit of normal (ULN) (patients \[Pts\] with Gilbert's syndrome are excluded from this requirement) * Serum glutamic oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]) and serum glutamate pyruvate transaminase (SGPT) (alanine aminotransferase \[ALT\]) =\< 3.0 x institutional upper limit of normal * Creatinine =\< 1.5 x ULN * Serum albumin \>= 2.8 g/dL * Lipase \< 2.0 x ULN and no radiologic or clinical evidence of pancreatitis * Urine protein/creatinine ratio (UPCR) =\< 1 * Serum phosphorus, calcium, magnesium and potassium \>= LLN * Prothrombin time (PT) \< 1.3 x ULN * International normalized ratio (INR) \< 1.3 x ULN * Partial thromboplastin time (PTT) test \< 1.3 x ULN * Women of childbearing potential must have a negative pregnancy test at screening; women of childbearing potential include women who have experienced menarche and who have not undergone successful surgical sterilization (hysterectomy, bilateral tubal ligation, or bilateral oophorectomy) or are not postmenopausal; postmenopause is defined as amenorrhea \>= 12 consecutive months; Note: 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, ovarian suppression or any other reversible reason * Women of child-bearing potential and men must agree to use adequate contraception; women of child-bearing potential and men must agree to use adequate contraception prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while she or her partner is participating in this study, she should inform her treating physician immediately; men treated or enrolled on this protocol must also agree to use adequate contraception prior to the study, for the duration of study participation, and 4 months after completion of cabozantinib administration; sexually active subjects (men and women) must agree to use medically accepted barrier methods of contraception (e.g., 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) * Ability to understand and the willingness to sign a written informed consent document Exclusion Criteria: * Prior treatment with cabozantinib or any tyrosine kinase inhibitor that targets MET or monoclonal antibody (mAb) targeting MET (such as onartuzumab \[MetMAb\]) * The subject has received radionuclide treatment =\< 6 weeks of the first dose of study treatment * The subject has received any other type of investigational agent =\< 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 active brain metastases or epidural disease; 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; baseline brain imaging with contrast-enhanced CT or MRI scans for subjects with known brain metastases is required to confirm eligibility; eligible patients with brain metastases can be taking anti-convulsant medications but only non-enzyme inducing anti-epileptic agents (NEIAED) will be allowed * 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 (e.g., clopidogrel); low dose aspirin (=\< 81 mg/day), low-dose warfarin (=\< 1 mg/day), and prophylactic low molecular weight heparin (LMWH) are permitted * The subject requires chronic concomitant treatment of strong cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) inducers (e.g., dexamethasone, phenytoin, carbamazepine, rifampin, rifabutin, rifapentin, phenobarbital, and St. John's Wort) or strong CYP3A4 inhibitors (ketoconazole, itraconazole, clarithromycin, indinavir, nefazodone, nelfinavir, ritonavir, voriconazole, and posaconazole); Note: because the lists of these agents are constantly changing, it is important to regularly consult a frequently-updated lists; 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 =\< 6 months before the first dose of study treatment * Hemoptysis of \>= 0.5 teaspoon (2.5 mL) of red blood =\< 3 months before the first dose of study treatment * Any other signs indicative of pulmonary hemorrhage =\< 3 months before the first dose of study treatment * The subject has radiographic evidence of cavitating pulmonary lesion(s) * The subject has tumor that is 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 =\< 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) \> 140 mmHg systolic, or \> 90 mmHg diastolic despite optimal antihypertensive treatment =\< 7 days of the first dose of study treatment * Any history of congenital long QT syndrome * Any of the following =\< 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 =\< 28 days before the first dose of study treatment * Intra-abdominal tumor/metastases invading GI mucosa * Active peptic ulcer disease as evidenced by esophagogastroduodenoscopy (EGD) * Inflammatory bowel disease (including ulcerative colitis and Crohn's disease), diverticulitis, cholecystitis, symptomatic cholangitis or appendicitis * Malabsorption syndrome * Any of the following =\< 6 months before the first dose of study treatment: * Abdominal fistula * Gastrointestinal perforation * Bowel obstruction or gastric outlet obstruction * 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 =\< 3 months before the first dose of study therapy * Other clinically significant disorders such as: * Active infection requiring systemic treatment =\< 28 days before the first dose of study treatment * Serious non-healing wound/ulcer/bone fracture =\< 28 days before the first dose of study treatment * History of organ transplant * History of major surgery as follows: * Major surgery =\< 3 months of the first dose of cabozantinib if there were no wound healing complications or =\< 6 months of the first dose of cabozantinib if there were wound complications * Minor surgery =\< 1 months of the first dose of cabozantinib if there were no wound healing complications or =\< 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 * The subject is unable to swallow tablets * The subject has a corrected QT interval calculated by the Fridericia or Bazett's formula (QTcF) \> 480 ms within 28 days before registration; Note: if initial QTcF or QTcB is found to be \> 480 ms, two additional electrocardiograms (EKGs) separated by at least 3 minutes should be performed; if the average of these three consecutive results for QTcF or QTcB is =\< 480 ms, the subject meets eligibility in this regard * 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 * History of allergic reactions attributed to compounds of similar chemical or biologic composition to cabozantinib * Any of the following: * Pregnant women * Nursing women * Men or women of childbearing potential who are unwilling to employ adequate contraception * Immunocompromised patients and patients known to be human immunodeficiency virus (HIV) positive and currently receiving antiretroviral therapy; Note: patients known to be HIV positive, but without clinical evidence of an immunocompromised state, are eligible for this trial * The subject has received prior treatment with a small molecule kinase inhibitor or a hormonal therapy (including investigation 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 the treatment