Sorafenib in Treating Patients With Refractory or Relapsed Acute Leukemia, Myelodysplastic Syndromes, or Blastic Phase Chronic Myelogenous Leukemia

Inclusion Criteria: * Diagnosis of 1 of the following: Acute myeloid leukemia (Acute promyelocytic leukemia (M3) allowed provided patient has failed prior therapy with both tretinoin and arsenic alone or in combination); Acute lymphoblastic leukemia; Myelodysplastic syndromes; Blastic phase chronic myelogenous leukemia (Failed OR intolerant to imatinib mesylate) * Must have failed prior therapy with \>= 1 cytotoxic- or biologic-targeted agent (e.g., hypomethylating agents, farnesyl transferase inhibitors, thalidomide, or tyrosine kinase inhibitors); Any number of prior regimens allowed * Performance status: ECOG 0-1 * ALT =\< 2.5 times upper limit of normal * Bilirubin =\< 1.5 mg/dL * Creatinine =\< 2.0 mg/dL OR Creatinine clearance \>= 60 mL/min * Fertile patients must use effective contraception * No psychiatric illness or social situation that would preclude study compliance * Prior bone marrow transplantation allowed * At least 2 weeks since prior cytotoxic agents OR at least 5 half-lives for non-cytotoxic agents in the absence of rapidly progressing disease * At least 24 hours since prior hydrea for control of peripheral blood leukemia cell counts * Hydroxyurea allowed up to 72 hours after start of therapy with sorafenib * No persistent, chronic, clinically significant toxicities \> grade 1 from prior chemotherapy Exclusion Criteria: * Cytopenias secondary to multilineage bone marrow failure allowed * Ineligible for or not willing to undergo allogeneic stem cell transplantation OR no donor available * Absolute blast count=\< 20,000/mm\^3 unless patient has documented fms-like tyrosine kinase 3 internal tandem duplication * No evidence of bleeding diathesis (except due to low platelets associated with the primary disease) * No New York Heart Association class III or IV congestive heart failure * No uncontrolled hypertension (i.e., sustained systolic blood pressure \[BP\] \>= 150 mm Hg or diastolic BP \>= 90 mm Hg) * No unstable angina pectoris * No symptomatic cardiac arrhythmia requiring and not responding to medical intervention * Not pregnant or nursing * No history of allergic reaction attributed to compounds of similar chemical or biological composition to the study drug * No swallowing dysfunction that would impede oral ingestion of tablets * No active uncontrolled infection * No other uncontrolled illness * No prior sorafenib * No other concurrent investigational or commercial agents, except for standard intrathecal chemotherapy for the treatment of isolated CNS leukemic involvement * No other concurrent anticancer agents * No concurrent combination antiretroviral therapy for HIV-positive patients * No concurrent therapeutic anticoagulation (Concurrent prophylactic anticoagulation \[i.e., low-dose warfarin, catheter flushing with heparin\] of venous or arterial access devices allowed) * No concurrent cytochrome P450 enzyme-inducing antiepileptic agents, including, but not limited to, any of the following: Phenytoin; Carbamazepine; Phenobarbital; Rifampin * No concurrent Hypericum perforatum (St. John's wort)