Pazopanib Hydrochloride in Treating Patients With Advanced Thyroid Cancer

Inclusion Criteria: * Histologically or cytologically confirmed differentiated, medullary or anaplastic thyroid cancer that is now advanced or metastatic; NOTE: patients with thyroid lymphomas or sarcomas are specifically excluded, as are patients with metastatic disease from other sites of origin to thyroid * Patients with confirmed differentiated thyroid cancer to be enrolled in the expanded/additional differentiated thyroid cancer (DTC) cohort must be thyroglobulin antibody negative * Zero, one or two prior therapeutic regimens (this includes cytotoxic plus non-cytotoxic therapeutic regimens) * Absence of sensitivity to therapeutic radioiodine (differentiated only) * Measurable disease, defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded) as \> 20 mm with conventional techniques or as \> 10 mm with spiral computed tomography (CT) scan; NOTE: disease that is measurable by physical examination only is not eligible * Life expectancy \> 3 months * Eastern Cooperative Oncology Group (ECOG) performance status (PS) 0, 1, or 2 (Karnofsky \>= 60%) * Leukocytes \> 3,000/mcL obtained =\< 7 days prior to registration * Absolute neutrophil count \> 1,500/mcL obtained =\< 7 days prior to registration * Platelets \> 100,000/mcL obtained =\< 7 days prior to registration * Total bilirubin =\< 1.5 X institutional upper limit of normal (ULN) obtained =\< 7 days prior to registration (if there is reason to believe that the patient has Gilbert's syndrome, the bilirubin can be fractionated; if the fractionated bilirubin is consistent with Gilbert's syndrome and there is no other possible explanation for the elevated indirect bilirubin, the patient may be eligible for the study if and only if the direct bilirubin is =\< 1.5 X institutional ULN) * Aspartate aminotransferase (AST) (serum glutamic oxaloacetic transaminase \[SGOT\]) \< 2.5 X institutional ULN obtained =\< 7 days prior to registration * Creatinine =\< 1.5 X ULN obtained =\< 7 days prior to registration * Proteinuria =\< + on urinalysis (may re-check) obtained =\< 7 days prior to registration * International normalized ratio (INR) =\< 1.2 X the ULN obtained =\< 7 days prior to registration * Blood pressure (BP) \< 140 mmHg (systolic) and \< 90 mmHg (diastolic); initiation or adjustment of BP medication is permitted prior to registration provided that the average of three BP readings at a visit prior to registration is \< 140/90 mmHg * Objective evidence of tumor progression in the 6 month period prior to GW786034 initiation as assessed by: * Unequivocal progression of objectively measured disease on successive appropriate imaging (e.g. CT scan); in cases of uncertainty of tumor progression, the principal investigator of the study will be available to assist in decisions * Women of child-bearing potential must have a negative serum pregnancy test =\< 7 days prior to registration; NOTE: women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for the duration of study participation; should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately; effective contraception is required for all fertile participants in the trial * Ability to understand and the willingness to sign a written informed consent document * Willingness to comply with the requirement of the study * Willingness to donate blood for correlative marker studies; (only applicable to sites within the United States) Exclusion Criteria: * Anaplastic, differentiated, medullary: a total of \> 2 prior therapeutic regimens (this total includes cytotoxic plus non-cytotoxic regimens); Note: enrollment of anaplastic, differentiated, and medullary patients who have had zero, one or two prior therapeutic regimens (cytotoxic plus non-cytotoxic regimens) is allowed - provided therapy ceased \> 21 days prior to registration; * NOTE: the principal investigator of the study should be contacted in the event of uncertainty related patient eligibility based upon prior therapies * Disease that is measurable by physical examination only * Any of the following: * Radiotherapy =\< 4 weeks prior to registration * Major surgery =\< 4 weeks prior to registration * Radiotherapy to \>= 25% of bone marrow * Concurrent therapy with octreotide unless tumor progression on this therapy has been demonstrated * Any other ongoing investigational agents * History of allergic reactions attributed to compounds of similar chemical or biologic composition to GW786034 (pazopanib) or other agents used in the study * \> +1 proteinuria (\< 30 mg/dL) on two consecutive dipstick or other urine assessments taken at least 1 week apart; NOTE: (in cases where questions arise related to disparate proteinuria measurements, the study principal investigator \[PI\] should be consulted for assistance in determining patient study eligibility) * Corrected QT interval (QTc) prolongation (defined as a QTc interval \>= 480 msecs) or other significant electrocardiogram (ECG) abnormalities (e.g. frequent ventricular ectopy, evidence of ongoing myocardial ischemia); NOTE: the principal investigator of the study should be contacted in the event of uncertainty related patient eligibility based upon ECG changes * Receiving cytochrome P450 (CYP) interactive concomitant medications; certain medications that act through the CYP450 system are specifically prohibited in patients receiving GW786034 (pazopanib) because in vitro data indicate that the agent has the potential to interact with the cytochrome P450 isoenzymes cytochrome P450, family 2, subfamily C, polypeptide 9 (CYP2C9) and cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4); certain other agents should be used with caution * Any condition (e.g., gastrointestinal tract disease resulting in an inability to take oral medication or a requirement for IV alimentation, prior surgical procedures affecting absorption, or active peptic ulcer disease) that impairs their ability to swallow and retain GSK786034 (pazopanib) * Any of the following conditions: * Serious or non-healing wound, ulcer, or bone fracture * History of abdominal fistula, gastrointestinal perforation, active diverticulitis, intra-abdominal abscess or gastrointestinal tract bleeding =\< 28 days of registration * Any history of cerebrovascular accident (CVA) =\< 6 months * Current use of therapeutic warfarin; Note: low molecular weight heparin and prophylactic low-dose warfarin (INR \< 1.2 X ULN) are permitted; prothrombin time (PT)/partial thromboplastin time (PTT) must meet the inclusion criteria * History of myocardial infarction, cardiac arrhythmia, admission for unstable angina, cardiac angioplasty or stenting within the last 12 weeks * History of venous thrombosis in last 12 weeks * Class III or IV heart failure as defined by the New York Heart Association (NYHA) functional classification system; NOTE: a patient who has a history of class II heart failure and is asymptomatic on treatment may be considered eligible * History of bleeding disorder, including patients afflicted with hemophilia, disseminated intravascular coagulation, or any other abnormality of coagulation potentially predisposing patients to bleeding * Poorly controlled depression or anxiety disorder, or recent (=\< 6 months) suicidal ideation * Known active and/or untreated brain metastases and/or brain metastases requiring ongoing therapy (e.g. corticosteroids); NOTE: (because of the poor prognosis often associated with brain metastases and because of the potential risk of bleeding in active brain metastases associated with multi-targeted tyrosine kinase inhibitor therapy, patients with active and/or untreated brain metastases and/or those with brain metastases requiring ongoing therapy - e.g. corticosteroids - are excluded from trial enrollment; enrollment will, however, be permitted in cases of patients with longstanding treated and inactive brain metastases not requiring ongoing therapy, providing that stability of brain metastases has been demonstrated for a period of 3 months or greater as assessed by intracranial imaging - and providing that there is no indication of increased vascularity of the treated metastases by magnetic resonance imaging (MRI) imaging conducted =\< 14 days prior to registration; when questions arise related to these criteria, the PI of the trial, Dr. Keith Bible, should be contacted for assistance on eligibility) * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection or psychiatric illness/social situations that would or might reasonably be expected to limit compliance with study requirements * Pregnant women; NOTE: (breastfeeding should be discontinued if the mother is treated with GW786034/pazopanib) * Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy; NOTE: (appropriate studies will be undertaken in patients receiving combination antiretroviral therapy when indicated) * Receiving any medications or substances known to affect or with the potential to affect the activity or pharmacokinetics of GW786034 (pazopanib); NOTE: the eligibility of patients will be determined following review of their case by the principal investigator; efforts should be made to switch patients who are taking enzyme-inducing anticonvulsant agents to other medications * Receiving any concomitant medications that are associated with a risk of QTc prolongation and/or Torsades de Pointes; NOTE: these medications should be discontinued or replaced with drugs that do not carry these risks, if possible