Iodine I 131 and Pazopanib Hydrochloride in Treating Patients With Recurrent and/or Metastatic Thyroid Cancer Previously Treated With Iodine I 131 That Cannot Be Removed By Surgery

Inclusion Criteria: * Subjects must provide written informed consent prior to performance of study-specific procedures or assessments, and must be willing to comply with treatment and follow up; procedures conducted as part of the subject's routine clinical management (e.g., blood count, imaging study) and obtained prior to signing of informed consent may be utilized for screening or baseline purposes provided these procedures are conducted as specified in the protocol * Histologically confirmed diagnosis of well-differentiated thyroid carcinoma (WDTC), including papillary and follicular subtypes, and documented recurrent and/or metastatic disease; patients must have unresectable disease: patients must not be amenable to surgery but prior thyroidectomy is allowed * Patient must have demonstrated evidence of disease progression by RECIST criteria using site assessment of computed tomography (CT)/magnetic resonance imaging (MRI) scans within 12 months (+1 month to allow for variances in patient scanning intervals) prior to study entry or by \> 50% increase in suppressed thyroglobulin levels during this time period * Patients with WDTC must be relatively 131I refractory/resistant as defined by at least one of the following: * One or more measurable lesions with low or absent 131I uptake on the most recent pre-study radioiodine scans, based on a visual review of scans or RAI scan reports * One or more measurable lesions with disease progression by RECIST within 12 months (+ 1 month to allow for variances in patient scanning intervals) of 131I therapy despite 131I uptake on RAI scan, based on site assessment of CT/MRI scans or by \> 50% increase in suppressed thyroglobulin levels during this time period * Evidence of at least one site of known disease with preserved 131I uptake above background levels on a diagnostic post-therapy 131I scan prior to study entry * Patients with WDTC must be receiving thyroxine suppression therapy and thyroid-stimulating hormone (TSH) should not be elevated (TSH should be =\< 5.50 mcu/mL) * Eastern Cooperative Oncology Group (ECOG) performance status of 0-2 * Absolute neutrophil count (ANC) \>= 1.5 X 10\^9/L * Hemoglobin \>= 9 g/dL (5.6 mmol/L) * Platelets \>= 90 X 10\^9/L * International normalized ratio (INR) =\< 1.2 X upper limit of normal (ULN); subjects receiving anticoagulant therapy are eligible if their INR is stable and within the recommended range for the desired level of anticoagulation * Activated partial thromboplastin time (aPTT) =\<1.2 X ULN * Total bilirubin =\< 1.5 X ULN * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) =\< 2.5 X ULN, and \< 5 X ULN in the presence of liver metastases; concomitant elevations in bilirubin and AST/ALT above 1.5 x ULN are not permitted * Serum creatinine =\< 2.0 mg/dL or, if serum creatinine \> 2.0 mg/dL, calculate creatinine clearance (CLCR) \>= 30 mL/min * Urine protein to creatinine ratio (UPC) \< 1 or, 24-hour urine protein \< 1 g; if UPC \>= 1, then a 24-hour urine protein must be assessed; subjects must have a 24-hour urine protein value \< 1 g to be eligible; use of urine dipstick for renal function assessment is not acceptable * Non-childbearing potential (i.e., physiologically incapable of becoming pregnant), including any female who has had: * A hysterectomy * A bilateral oophorectomy (ovariectomy); a bilateral tubal ligation * Is post-menopausal (subjects not using hormone replacement therapy \[HRT\] must have experienced total cessation of menses for \>= 1 year and be greater than 45 years in age, OR, in questionable cases, have a follicle stimulating hormone \[FSH\] value \> 40 mIU/mL and an estradiol value \< 40pg/mL \[\< 140 pmol/L\]; subjects using HRT must have experienced total cessation of menses for \>= 1 year and be greater than 45 years of age OR have had documented evidence of menopause based on FSH and estradiol concentrations prior to initiation of HRT) * Childbearing potential, including any female who has had a negative serum pregnancy test within 2 weeks prior to the first dose of study treatment, preferably as close to the first dose as possible, and agrees to use adequate contraception for at least 2 weeks following the last dose of the investigational product * GlaxoSmithKline (GSK) acceptable contraceptive methods, when used consistently and in accordance with both the product label and the instructions of the physician, are as follow: * Complete abstinence from sexual intercourse for 14 days before exposure to investigational product, through the dosing period, and for at least 21 days after the last dose of investigational product * Oral contraceptive, either combined or progestogen alone * Injectable progestogen * Implants of levonorgestrel * Estrogenic vaginal ring * Percutaneous contraceptive patches * Intrauterine device (IUD) or intrauterine system (IUS) with a documented failure rate of less than 1% per year * Male partner sterilization (vasectomy with documentation of azoospermia) prior to the female subject's entry into the study, and this male is the sole partner for that subject * Double barrier method: condom and an occlusive cap (diaphragm or cervical/vault caps) with a vaginal spermicidal agent (foam/gel/film/cream/suppository) * Female subjects who are lactating should discontinue nursing prior to the first dose of study drug and should refrain from nursing throughout the treatment period and for 14 days following the last dose of study drug Exclusion Criteria: * Patients with medullary thyroid cancer, thyroid lymphoma or anaplastic thyroid cancer are excluded * Resolution of all acute toxic effects of prior systemic therapy (including iodine therapy or systemic therapy), radiotherapy or surgical procedure to National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE) version 4.0 grade =\< 1 * Patients with cumulative iodine I 131 exposure in excess of 1000 mCi * Second primary malignancy that is of clinical significance, clinical detectable and/or progressing at the time of consideration for study enrollment * History or clinical evidence of central nervous system (CNS) metastases or leptomeningeal carcinomatosis, except for individuals who have previously-treated CNS metastases, are asymptomatic, and have had no requirement for steroids for 28 days prior to first dose of study drug; screening with CNS imaging studies (CT or MRI) is required only if clinically indicated or if the subject has a history of CNS metastases; clinically significant gastrointestinal abnormalities that may increase the risk for gastrointestinal bleeding including, but not limited to: * Active peptic ulcer disease * Known intraluminal metastatic lesion/s with risk of bleeding * Inflammatory bowel disease (e.g. ulcerative colitis, Crohn's disease), or other gastrointestinal conditions with increased risk of perforation * History of abdominal fistula, gastrointestinal perforation, or intra abdominal abscess within 28 days prior to beginning study treatment * Clinically significant gastrointestinal abnormalities that may affect absorption of investigational product including, but not limited to: * Malabsorption syndrome * Major resection of the stomach or small bowel * Presence of uncontrolled infection * Corrected QT interval (QTc) \> 480 msecs using Bazett's formula * History of any one or more of the following cardiovascular conditions within the past 6 months: * Cardiac angioplasty or stenting * Myocardial infarction * Unstable angina * Coronary artery bypass graft surgery * Symptomatic peripheral vascular disease * Class III or IV congestive heart failure, as defined by the New York Heart Association (NYHA) * Poorly controlled hypertension \[defined as systolic blood pressure (SBP) of \>= 140 mmHg or diastolic blood pressure (DBP) of \>= 90mmHg\] * Note: initiation or adjustment of antihypertensive medication(s) is permitted prior to study entry * History of cerebrovascular accident including transient ischemic attack (TIA), pulmonary embolism or untreated deep venous thrombosis (DVT) within the past 6 months; subjects with recent DVT who have been treated with therapeutic anti-coagulating agents for at least 6 weeks are eligible * Prior major surgery or trauma within 28 days prior to first dose of study drug and/or presence of any non-healing wound, fracture, or ulcer (procedures such as catheter placement not considered to be major) * Evidence of active bleeding or bleeding diathesis * Known endobronchial lesions and/or lesions infiltrating major pulmonary vessels that increase the risk of pulmonary hemorrhage; Note: lesions infiltrating major pulmonary vessels (contiguous tumor and vessels) are excluded; however, the presence of a tumor that is touching, but not infiltrating (abutting) the vessels is acceptable (CT with contrast is strongly recommended to evaluate such lesions) * Recent Hemoptysis in excess of 15 ml of bright red blood in the 8 weeks prior to study entry * Any serious and/or unstable pre-existing medical, psychiatric, or other condition that could interfere with subject's safety, provision of informed consent, or compliance to study procedures * Unable or unwilling to discontinue use of prohibited medications for at least 14 days or five half-lives of a drug (whichever is longer) prior to the first dose of study drug and for the duration of the study * Treatment with any of the following anti-cancer therapies: * Radiation therapy, surgery or tumor embolization within 14 days prior to the first dose of pazopanib OR * Chemotherapy, immunotherapy, biologic therapy, investigational therapy or hormonal therapy within 14 days or five half-lives of a drug (whichever is longer) prior to the first dose of pazopanib