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)