Intensity-Modulated Radiation Therapy and Paclitaxel With or Without Pazopanib Hydrochloride in Treating Patients With Anaplastic Thyroid Cancer

Inclusion Criteria: * Pathologically (histologically or cytologically) proven diagnosis of anaplastic thyroid cancer (a diagnosis that is noted to be "consistent with anaplastic thyroid cancer" with the presence of a thyroid mass is acceptable) * Note: Tissue collection for central review is mandatory, but central review is not required for eligibility; due to the aggressiveness of this disease, treatment will be started prior to central review * If there was a total or partial thyroidectomy completed within 3 months of enrollment, the surgical specimen must show the area of anaplastic thyroid cancer to be at least 1 cm in greatest dimension * The following minimum diagnostic workup is required: * History/physical examination within 2 weeks prior to registration * Imaging of neck and brain (computed tomography \[CT\] scan or magnetic resonance imaging \[MRI\]) and chest/abdominal imaging (chest x-ray or chest CT scan, or full body positron emission tomography \[PET\]/CT are acceptable) within 4 weeks prior to registration * Note: The CT scan of the neck must be done with contrast or if an MRI is done, with gadolinium; therefore, the CT portion of a full body PET/CT has to be a high resolution CT to be acceptable for eligibility * Abdominal imaging must cover the liver and adrenal glands; therefore, separate imaging is not required if these areas are covered by a chest CT scan * Electrocardiogram within 10 days prior to registration * Zubrod performance status 0-2 * Absolute neutrophil count (ANC) \>= 1,500 cells/mm\^3 (within 10 days prior to registration on study) * Platelets \>= 100,000 cells/mm\^3 (within 10 days prior to registration on study) * Hemoglobin (Hgb) \>= 9.0 g/dl (within 10 days prior to registration on study) (Note: the use of transfusion or other intervention to achieve Hgb \>= 9.0 g/dL is acceptable) * Total bilirubin \< 1.5 x institutional upper limit of normal (ULN) (except for patients with Gilbert's syndrome and elevations of indirect bilirubin) (within 10 days prior to registration) * Aspartate aminotransferase (AST) or alanine aminotransferase (ALT) \< 2.5 x institutional ULN (within 10 days prior to registration); Note: patients who have both bilirubin \> ULN and AST/ALT \> ULN are not eligible (unless they have Gilbert's syndrome and elevations of indirect bilirubin) * Spot urine protein to creatinine ratio (UPCR) \< 1 or a 24-hour urine protein collection \< 1 gm within 10 days prior to registration * Creatinine \< 1.5 mg/dL or within normal institutional limits within 10 days prior to registration; Note: if neither criteria is met, the creatinine clearance must be \> 50 mL/min/1.73 m\^2 per either the Cockcroft-Gault equation, Jeliffe method, or 12- or 24-hour urine collection * Serum electrolytes including sodium, potassium, blood urea nitrogen (BUN), creatinine, glucose, magnesium, phosphate, and calcium within 10 days prior to registration * Documentation of the patient's history of corrected QT interval (QTc) prolongation, family history of prolonged QTc, and relevant cardiac disease within 10 days prior to registration * Evaluation of the patient's medications within 10 days prior to registration with attempt to change any medication that affects cytochrome P450, family 3, subfamily A, polypeptide 4 (CYP3A4) * Blood pressure =\< 140/90 within 10 days of registration (must be taken and recorded by a health care professional); Note: if the systolic blood pressure is \> 140 and/or diastolic blood pressure is \> 90 at the time of registration, the patient's blood pressure must be controlled; systolic blood pressure must be \< 140 and diastolic blood pressure must be \< 90 on at least 2 separate measurements prior to the start of treatment, and the treating physician must believe that this is feasible in order to enroll the patient * Prothrombin time (PT)/international normalized ratio (INR)/partial thromboplastin time (PTT) within 1.2 x the upper limit of normal within 10 days prior to registration unless the patient is receiving coumadin and has a stable INR that is in range for the desired level of anticoagulation * Negative pregnancy test (serum or urine) within 10 days of registration in women of child-bearing potential * Women of childbearing potential and male participants who are sexually active must agree to practice adequate contraception during treatment and for 6 months post-treatment * The patient must provide study specific informed consent prior to study entry Exclusion Criteria: * Known active invasive malignancy (except for non-melanomatous skin cancer or anaplastic thyroid cancer; the presence of prostate cancer confined to the prostate with a prostate-specific antigen \[PSA\] =\< 1 ng/mL for more than 6 months also is allowed) * Prior systemic chemotherapy for anaplastic thyroid cancer * Patients who have had chemotherapy or radiotherapy within 4 weeks of registration (6 weeks for nitrosoureas or mitomycin C) prior to entering the study or those who have not recovered from adverse events due to agents administered \> 4 weeks previously * Patients receiving other investigational agents * Prior radiotherapy to the region of the study cancer that would result in overlap of radiation therapy fields * Patients with any of the following cardiovascular conditions within the past 6 months: * Cerebrovascular accident (CVA) or transient ischemic attack (TIA) * Admission for unstable angina * Myocardial Infarction * Cardiac angioplasty or stenting * Coronary artery bypass graft surgery * Pulmonary embolism, untreated deep venous thrombosis (DVT), or DVT which has been treated with therapeutic anticoagulation for less than 6 weeks * Arterial thrombosis * Symptomatic peripheral vascular disease * 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 III heart failure and is asymptomatic on treatment may be considered eligible for the study * Certain medications that are associated with a risk for QTc prolongation and/or torsades de pointes, although not prohibited, should be avoided or replaced with medications that do not carry these risks, if possible * Patients who require heparin (other than low-molecular weight heparin) * Patients with any condition that may impair the ability to absorb oral medications/investigational product including: * Prior surgical procedures affecting absorption including, but not limited to, major resection of stomach or small bowel * Active peptic ulcer disease * Malabsorption syndrome * Patients with any condition that may increase the risk of gastrointestinal bleeding or gastrointestinal perforation, including: * Active peptic ulcer disease * Known intraluminal metastatic lesions * Inflammatory bowel disease (e.g., ulcerative colitis, Crohn's disease) or other gastrointestinal conditions which increase the risk of perforation * History of abdominal fistula, gastrointestinal perforation, or intra-abdominal abscess within 28 days prior to beginning study treatment * History of hemoptysis within 30 days of registration; Note: patients who have minimal bleeding from the mouth, which is clearly not related to a source in the lungs, i.e., surgery such as a non-lung biopsy, are eligible only after good hemostasis has been documented * Uncontrolled intercurrent illness including, but not limited to, ongoing or active infection or psychiatric illness/social situations that would limit compliance with study requirements * Pregnancy or women of childbearing potential and men who are sexually active and not willing/able to use medically acceptable forms of contraception; this exclusion is necessary because the treatment involved in this study may be significantly teratogenic * Prior allergic reaction to the study drug(s) involved in this protocol * QTc prolongation defined as a QTc interval \>= 480 msecs or other significant electrocardiogram (EKG) abnormalities are ineligible; Note: if unsure about EKG abnormality, the treating physician should discuss this with Drs. Sherman or Bible * Known brain metastasis * Human immunodeficiency virus (HIV)-positive patients on combination antiretroviral therapy because of the potential for pharmacokinetic interactions with pazopanib; in addition, these patients are at increased risk of lethal infections when treated with marrow-suppressive therapy; appropriate studies will be undertaken in patients receiving combination antiretroviral therapy when indicated * Certain medications that act through the cytochrome P450 (CYP450) system are specifically prohibited in patients receiving pazopanib and others should be avoided or administered with extreme caution * Strong inhibitors of CYP3A4 such as ketoconazole, itraconazole, clarithromycin, atazanavir, indinavir, nefazodone, nelfinavir, ritonavir, saquinavir, telithromycin, voriconazole may increase pazopanib concentrations and are prohibited; although, in exceptional circumstances, they may be administered in conjunction with lowering the dose of pazopanib by 50% of what would otherwise be administered; grapefruit juice is also an inhibitor of CYP450 and should not be taken with pazopanib * Strong inducers of CYP3A4, such as rifampin, may decrease pazopanib concentrations, are strictly prohibited * Medications that have narrow therapeutic windows and are substrates of CYP3A4, cytochrome P450, family 2, subfamily D, polypeptide 6 (CYP2D6), or cytochrome P450, family 2, subfamily C, polypeptide 8 (CYP2C8) should be avoided and, if necessary, administered with caution