Inolitazone Dihydrochloride and Paclitaxel in Treating Patients With Advanced Anaplastic Thyroid Cancer

* Patients must have histologically or cytologically diagnosed advanced anaplastic thyroid cancer (ATC)
* Patients must have measurable disease
* Patients must have either metastatic (stage IVC) or locally advanced unresectable disease (stage IVB)
* Patients should have resolution of any toxic effects of prior therapy (except alopecia) to National Cancer Institute (NCI) Common Terminology Criteria for Adverse Events (CTCAE), version 4.0, grade 1
* There is no limit to the number of prior lines of treatment a patient has received
* No treatment with chemotherapy, radiation therapy, immunotherapy, biological therapy, hormonal therapy, or other thiazolidinediones (TZDs) =\< 21 days before study registration
* No prior taxane therapy =\< 6 months, except as a radiosensitizer
* No history of the following:

  * Class III or IV congestive heart failure (CHF)
  * Grade 3 or 4 thromboembolic event =\< 6 months
  * Pericardial effusion =\< 12 months (any grade)
  * Pericardial involvement with tumor
  * Grade 2 or higher pleural effusion =\< 6 months
* No current symptomatic, untreated, or uncontrolled brain metastases present
* No major surgery =\< 14 days prior to registration
* No grade 2 or higher neuropathy
* No known history of severe hypersensitivity reactions to any of the components of efatutazone or paclitaxel formulations
* Not pregnant and not nursing; women of childbearing potential only, a negative pregnancy test done =\< 7 days prior to registration is required
* Patients with diabetes mellitus requiring concurrent treatment with insulin or thiazolidinedione (TZD) oral agents are not eligible
* Patients with known hypersensitivity to any TZD oral agents are not eligible
* Eastern Cooperative Oncology Group (ECOG) performance status =\< 2
* Absolute neutrophil count (ANC) \>= 1,500/mm\^3
* Platelet count \>= 100,000/mm\^3
* Creatinine =\< 1.5 x upper limit of normal (ULN) mg/dL OR calculated (calc.) creatinine clearance \>= 60 mL/min
* Bilirubin =\< 1.5 x ULN
* Aspartate aminotransferase (AST) =\< 2.5 x ULN
* Although they will not be considered formal eligibility (exclusion) criteria, physicians should recognize that the following may seriously increase the risk to the patient entering this protocol:

  * Psychiatric illness which would prevent the patient from giving informed consent
  * Medical condition such as uncontrolled infection (including human immunodeficiency virus \[HIV\]), uncontrolled diabetes mellitus or cardiac disease which, in the opinion of the treating physician, would make this protocol unreasonably hazardous for the patient
  * Patients with a "currently active" second malignancy other than non-melanoma skin cancers; patients are not considered to have a "currently active" malignancy if they have completed therapy and are free of disease for \>= 3 years; there is an exception for patients with a history of well differentiated thyroid cancer that has progressed to anaplastic thyroid cancer
  * Patients who cannot swallow oral formulations of the agent(s)
  * Women and men of reproductive potential should agree to use an appropriate method of birth control throughout their participation in this study ; appropriate methods of birth control include abstinence, oral contraceptives, implantable hormonal contraceptives or double barrier method (diaphragm plus condom)
  * Efatutazone is metabolized by cytochrome P450, family 3, subfamily A, polypeptide 4/5 (CYP3A4/5), and inhibits CYP2C8, 2C9, 2C19, and 3A4, and is a substrate of P-glycoprotein (PgP) and breast cancer resistance protein (BCRP)