Azacitidine and Entinostat in Treating Patients With Advanced Breast Cancer

Inclusion Criteria: * Patient must have histologically or cytologically confirmed adenocarcinoma of the breast triple-negative (ER-, progesterone receptor \[PR\]-, human epidermal growth factor receptor 2 \[HER2\]-) or hormone positive/ HER2-, with evidence of locally advanced and inoperable or metastatic disease (American Joint Committee on Cancer \[AJCC\] Stage IV) * NOTE: Triple-negative patients will be defined per American Society of Clinical Oncology-College of American Pathologists (ASCO-CAP) Guidelines; these guidelines state that ER and PR assays be considered positive if there are at least 1% positive tumor nuclei in the sample on testing in the presence of expected reactivity of internal (normal epithelial elements) and external controls * A patient who has a change in receptor status (e.g., PR negative to positive) may be stratified as triple negative or hormone positive, contrary to the most recent receptor testing, for the purposes of the study based upon the clinical course at the discretion of the Study Chair; for HER2 assessment, a negative result is an immuno-histochemistry staining of 0 or 1+, a fluorescence in situ hybridization (FISH) result of less than 4.0 HER2 gene copies per nucleus, or a FISH ratio of less than 1.8 * Patients with triple negative disease must have progressed through at least 1 prior chemotherapy regimen (administered in the adjuvant or metastatic setting); hormone receptor-positive patients must have progressed through two lines of hormonal therapy (administered in the adjuvant or metastatic setting), unless otherwise eligible as per below, and at least 1 prior chemotherapy regimen (administered in the adjuvant or metastatic setting) with no known curative options available * NOTE: Patients with hormone receptor-positive disease may be considered eligible if deemed clinically hormone-resistant taking into consideration the rate of progression of disease or a short interval of time on first line hormonal therapy before progression, or if intolerant of hormonal therapy such that further hormonal therapy will not be considered as part of the treatment strategy * In patients with metastatic disease in the liver, liver disease burden is limited to no more than 30% of total liver volume as assessed by local review * Patients must have measurable disease * Life expectancy of \>= 12 weeks * Eastern Cooperative Oncology Group (ECOG) performance status 0 or 1 * Hemoglobin (HgB) \>= 9.0 g/dL * Absolute neutrophil count (ANC) \>= 1,500/mcL * Platelet count \>= 100,000/mcL * Total bilirubin =\< 1.5 x institutional upper limit of normal (ULN); an exception to this may be allowed for participants with Gilbert's syndrome with documented approval by the Protocol Chair * Serum glutamic oxaloacetic transaminase (SGOT) (aspartate aminotransferase \[AST\]) =\< 3 x ULN * Creatinine =\< institutional ULN or creatinine clearance \>= 60 mL/min using the Modified Cockcroft-Gault formula * Negative (serum or urine) pregnancy test done =\< 7 days prior to registration, for women of childbearing potential only * Patient must have an accessible tumor lesion from which a biopsy specimen can be obtained; NOTE: if baseline biopsy is attempted and is unsuccessful (eg, patient intolerance, inadequate tissue), the patient will still be considered eligible for the study * Ability to understand and the willingness to sign a written informed consent document * Willingness to provide tissue and blood samples for mandatory translational research * Willingness to return to the enrolling institution for follow-up Exclusion Criteria: * Any of the following: * Pregnant women * Nursing women * Men or women of childbearing potential who are unwilling to employ adequate contraception * NOTE: should a woman become pregnant or suspect she is pregnant while participating in this study, she should inform her treating physician immediately * Any of the following: * Chemotherapy \< 3 weeks prior to registration * Hormone therapy \< 3 weeks prior to registration * Radiotherapy \< 3 weeks prior to registration * Surgery \< 3 weeks prior to registration * Nitrosoureas/mitomycin C \< 6 weeks prior to registration * Trastuzumab \< 6 weeks prior to registration * Bevacizumab \< 6 weeks prior to registration * Those who have not recovered from acute adverse events to grade \< 2 or baseline due to agents administered, with exception of alopecia, unless approved by the Protocol Chair * NOTE: concurrent bisphosphonate therapy is allowed; concurrent ovarian suppression therapy (i.e., Lupron or Zoladex) is also allowed at the discretion of the Protocol Chair/designee * Any other ongoing investigational agents * Known sensitivity to 5-AZA, entinostat or mannitol * Uncontrolled intercurrent illness that in the judgment of the investigator, would make the patient inappropriate for entry into this study or interfere significantly with the proper assessment of safety and toxicity of the prescribed regimens including, but not limited to: * Ongoing or active infection * Symptomatic congestive heart failure (New York Heart Association \[NYHA\] class II or above) * Unstable angina pectoris * Cardiac arrhythmia * Psychiatric illness/social situations that would limit compliance with study requirements * Other co-morbid systemic illness or other severe concurrent disease * Active malignancy other than breast cancer =\< 3 years prior to registration; EXCEPTIONS: non-melanotic skin cancer or carcinoma-in-situ of the cervix; NOTE: if there is a history of prior malignancy, they must not be receiving other specific treatment for their cancer * Immunocompromised patients (other than that related to the use of corticosteroids) including patients known to be human immunodeficiency virus (HIV) positive * Received prior treatment with HDAC (histone deacetylase) inhibitors or demethylating agents =\< 2 weeks prior to registration * Unstable brain metastases; NOTE: patients with brain metastases must have stable neurologic status and magnetic resonance imaging (MRI) imaging following local therapy (surgery or radiation) for at least 4 weeks, with no dexamethasone requirement (stable low dose dexamethasone allowed at discretion of Study Chair); patients with leptomeningeal disease are not eligible * Patient taking valproic acid * Patient who cannot swallow tablets