Entinostat and Sorafenib Tosylate in Treating Patients With Advanced or Metastatic Solid Tumors or Refractory or Relapsed Acute Myeloid Leukemia

Inclusion Criteria: * Must meet 1 of the following criteria: * Histologically or cytologically confirmed solid tumors (dose-escalation only) * Locally advanced, inoperable, or metastatic disease * Evaluable or measurable disease * Diagnosis of acute myeloid leukemia (AML) for which no other standard therapy, including stem cell transplantation, is expected to result in meaningful clinical response (expansion cohort only) * Refractory or relapsed disease * Chronic myelogenous leukemia in blast crisis allowed * No acute promyelocytic leukemia with t(15;17) * Must consent to have fresh tumor, bone marrow aspirate, and biopsy obtained * No untreated, symptomatic, or unstable brain metastases * No active CNS involvement for patients with AML * ECOG performance status 0-1 * ANC ≥ 1,500/mm³ (dose-escalation only) * Platelet count ≥ 100,000/mm³ (dose-escalation only) * Hemoglobin ≥ 10 g/dL (dose-escalation only) * Total bilirubin ≤ 1.5 times upper limit of normal (ULN) * ALT and AST ≤ 3 times ULN (≤ 5 times ULN for liver metastasis) * Creatinine clearance ≥ 40 mL/min * Albumin \> 3.0 g/dL * Plasma phosphorus \> lower limit of normal (with supplementation) * INR ≤ 1.5 * APTT ≤ 1.5 times ULN (if not on anticoagulants) * Able to swallow oral medications * ≥ 16 years old (expanded cohort patients recruited at the University of Colorado site) * Must have tolerated prior sorafenib tosylate (dose: 400 mg twice daily), if applicable * Not pregnant or nursing * Negative pregnancy test * Fertile patients must use effective double-barrier contraception during the study and for 90 days after completion of study therapy * No history of cardiac disease, including any of the following: * NYHA class II-IV congestive heart failure * Active coronary artery disease * Prior diagnosis of bradycardia or other cardiac arrhythmia defined as ≥ grade 2 or uncontrolled hypertension * Myocardial infarction (MI) within the past 6 months * Persistent tachycardia * LVEF \< 40% by MUGA * Second- or third-degree heart block * QTc \> 490 msec * ST-T wave changes consistent with acute MI or acute ischemia * No clinically active serious infections defined as ≥ grade 2 * No substance abuse, medical, psychological, or social conditions that may, in the opinion of the Investigator, interfere with the patient's participation or evaluation of the study results * No condition that is unstable or that could jeopardize the safety of the patient and his/her study compliance * No known HIV infection * No significant gastrointestinal disorder that, in the opinion of the Investigator, could interfere with the absorption of entinostat and/or sorafenib tosylate including any of the following: * Significant, uncontrolled inflammatory bowel disease * Abdominal fistula or gastrointestinal perforation within the past 6 months * Extensive small bowel resection * Requiring tube feeding or parenteral hydration and/or nutrition * No concurrent immunosuppressive therapies, including high-dose systemic corticosteroids (\> 0.5 mg/kg/day prednisone or equivalent) unless used intermittently or as a tapered course for ≤ 4 weeks * Concurrent hydroxyurea and/or anagrelide allowed * Concurrent warfarin allowed provided the dose has been stable for the past 2 months and INR has been between 2 and 3 * More than 4 weeks since prior chemotherapy, immunotherapy, or investigational agents (=\< 2 weeks for leukemia patients in expansion cohort) * More than 2 weeks since prior palliative radiotherapy * More than 4 weeks since major surgery * More than 2 weeks since minor surgery (e.g., talc pleurodesis, excisional biopsy, etc.) * Concurrent hormonal therapies (e.g., LHRH antagonists, megestrol, octreotide, calcitonin, etc.) allowed * No concurrent strong CYP3A4 inducers or inhibitors, including, but not limited to, any of the following: * Valproic acid * Rifampin * Phenobarbital * Phenytoin * Carbamazepine * Ketoconazole * Erythromycin * Grapefruit * No other concurrent anticancer therapy including chemotherapy, radiotherapy (including palliative), or immunotherapy (except hydroxyurea in leukemia patients during course 1) * No other concurrent investigational agents