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 …