Entinostat and Clofarabine in Treating Patients With Newly Diagnosed, Relapsed, or Refractory Poor-Risk Acute Lymphoblastic Leukemia or Bilineage/Biphenotypic Leukemia

Inclusion Criteria: * Adults age \>= 40 years with the established, pathologically-confirmed diagnoses of newly diagnosed ALL or ABL are eligible for study; adults with relapsed and refractory ALL or ABL who are \>= 21 years of age and who have progressive disease following their last therapy are eligible for study; the additional following criteria must be met: * For adults with relapsed/refractory ALL, no more than 5 previous regimens * Eastern Cooperative Oncology Group (ECOG) performance status 0, 1, or 2 * Patients must be able to give informed consent * Female patients of childbearing age must have negative pregnancy test * Total white blood cell count (WBC) =\< 150,000 with no evidence for ongoing or impending leukostasis * Total bilirubin =\< 2.0 mg/dL unless elevated due to Gilbert's disease, hemolysis or leukemic infiltration * Aspartate aminotransferase (AST)/alanine aminotransferase (ALT) =\< 2.5 × upper limit of normal (ULN) unless due to leukemic infiltration * Serum creatinine =\< 2.0 mg/dL * Left ventricular ejection fraction (LVEF) \>= 45% as measured by echocardiogram (ECHO) or multi gated acquisition (MUGA) scan * Women of child-bearing potential and men must agree to use adequate contraception (hormonal or barrier method of birth control; abstinence) prior to study entry and for 30 days after study participation; should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately * Patients who have undergone stem cell transplantation (SCT), autologous or allogeneic, are eligible provided that they are \>= 4 weeks from stem cell infusion, have no active graft-vs-host disease (GVHD), and meet other eligibility criteria * Patients who fail primary induction therapy or relapse after achieving complete remission (CR) are eligible if they have undergone no more than 5 prior cytotoxic regimen, \>= 14 days off cytotoxic chemotherapy, and \>= 2 weeks radiation therapy; patients must be off biologic therapies \>= 7 days, must be off hematopoietic growth factors \>= 3 days; if using hydroxyurea steroids, imatinib, arsenic, interferon, or other non-cytotoxics for blast count control, patient must be off for \>= 24 hrs before starting entinostat plus clofarabine Exclusion Criteria: * Philadelphia chromosome positive ALL * Patients may not have received previous treatment with entinostat or other histone deacetylase (HDAC) inhibitors (including valproic acid) or clofarabine within the previous 6 months * Concomitant chemotherapy, radiation therapy, or immunotherapy * Hyperleukocytosis with \>= 150,000 blasts/uL (if using hydroxyurea, steroids, maintenance doses of 6-mercaptopurine and/or methotrexate, arsenic, interferon or leukapheresis for blast count control, patient must be off those agents for 24 hours prior to beginning entinostat plus clofarabine) * Active disseminated intravascular coagulation (DIC) * Active central nervous system (CNS) leukemia; patients with known previous CNS leukemia may continue to receive intrathecal therapy with cytarabine (ara-C), methotrexate, and/or thiotepa plus steroids as prophylaxis against reactivation of active CNS disease * Use of investigational cytotoxic agents within 30 days or any anticancer therapy within 14 days before study entry, except for hydroxyurea and steroids, both of which must be discontinued at least 24 hours before study entry; the patient must have recovered from all acute toxicities from any previous therapy; the patient must have recovered from all acute toxicities from any previous therapy * Patients must have discontinued all growth factors at least 3 days before study * History of severe coronary artery disease, including myocardial infarction within the previous 3 months, arrhythmias other than atrial flutter or fibrillation requiring medication, or uncontrolled congestive heart failure * Dyspnea at rest or with minimal exertion * Active uncontrolled infection; patients with infection under active treatment and controlled with antibiotics are eligible * Patients with active \>= grade 2 graft versus host disease (GVHD) * Presence of other life-threatening illness * Patients with mental deficits and/or psychiatric history that preclude them from giving informed consent or from following protocol * Pregnant or nursing women; breastfeeding should be discontinued if the mother is treated with entinostat * Male and female patients who are fertile who do not agree to use an effective barrier method of birth control (i.e., hormonal or barrier method of birth control; abstinence) to avoid pregnancy during the study and for a minimum of 30 days after study treatment * Previous history of or current seizure disorder * Human immunodeficiency virus (HIV) infected patients who have cluster of differentiation (CD4) cell count =\< 350/mm\^3 or a history of acquired immunodeficiency syndrome (AIDS)-defining conditions