Sorafenib Tosylate in Treating Patients With Locally Advanced, Metastatic, or Locally Recurrent Thyroid Cancer

You may be eligible if

• Histologically confirmed diagnosis of 1 of the following:
• * Papillary thyroid cancer (stratum I)
• * Papillary, follicular, Hurthle cell, insular, or anaplastic thyroid cancer (stratum II)
• * Mixed histology, poorly differentiated, or tall-cell variants allowed
• Metastatic, locally advanced, or locally recurrent disease
• At least 1 unidimensionally measurable lesion \>= 20 mm by conventional techniques OR \>= 10 mm by spiral CT scan
• * The following are considered non-measurable disease:
• * Tumors in a previously irradiated area
• * Bone lesions
• * Leptomeningeal disease
• * Ascites
• * Pleural/pericardial effusion
• * Lymphangitis cutis/pulmonis
• * Abdominal masses not confirmed and followed by imaging techniques
• * Cystic lesions
• Archival tumor tissue block OR material collected before study entry available (stratum I)
• Biopsy-accessible disease (stratum I)
• Performance status - ECOG 0-1
• At least 6 months
• WBC \>= 3,000/mm\^3
• Absolute neutrophil count \>= 1,500/mm\^3
• Platelet count \>= 100,000/mm\^3
• No bleeding diathesis
• Bilirubin =\< 1.5 times upper limit of normal (ULN)
• AST and ALT =\< 1.5 times ULN
• Creatinine =\< 1.5 times ULN
• No uncontrolled hypertension
• Not pregnant or nursing
• Negative pregnancy test
• Fertile patients must use effective contraception
• Willing to undergo 2 tumor biopsies during study participation (stratum I)
• No history of allergic reaction attributed to compounds of similar chemical or biologic composition to sorafenib
• No ongoing or active infection
• No psychiatric illness or social situation that would preclude study compliance
• No other uncontrolled illness
• No other concurrent malignancy except nonmetastatic nonmelanoma skin cancer or carcinoma in situ of the cervix
• No prior systemic chemotherapy for thyroid cancer (stratum I)
• * Prior systemic chemotherapy used to treat a second primary cancer with curative intent allowed provided the primary cancer was treated more than 5 years before study entry
• No more than 3 prior systemic chemotherapy regimens for thyroid cancer (stratum II)
• More than 6 weeks since prior systemic chemotherapy (stratum II)
• No prior external beam radiotherapy to the sole site of measurable disease (except for patients with anaplastic thyroid cancer)
• More than 6 weeks since prior external beam radiotherapy
• More than 24 weeks since prior iodine I 131
• Recovered from all prior therapy
• No prior sorafenib
• More than 6 weeks since prior investigational tumor-specific therapy
• Concurrent oral or IV bisphosphonates for bony metastases allowed at the discretion of the investigator
• No other concurrent tumor-specific or investigational therapy
• No concurrent combination antiretroviral therapy for HIV-positive patients
• No concurrent therapeutic anticoagulation
• * Concurrent prophylactic anticoagulation (e.g., low-dose warfarin) for venous or arterial access devices allowed provided PT, INR, or PTT are normal