Exemestane With or Without Entinostat in Treating Patients With Recurrent Hormone Receptor-Positive Breast Cancer That is Locally Advanced or Metastatic

Inclusion Criteria: * Estrogen receptor (ER) and/or progesterone receptor (PR) positive histologically confirmed adenocarcinoma of the breast with staining of \>= 1% cells will be considered positive; receptor status may be based on any time during treatment prior to study randomization, and from any site (i.e. primary, recurrent, or metastatic) * Patients whose tumors have HER2 immunohistochemistry (IHC) 3+, in situ hybridization (ISH) \>= 2.0, or average HER2 copy number \>= 6.0 signals per cell are not eligible; receptor status may be based on any time during treatment prior to study randomization, and from any site (i.e. primary, recurrent, or metastatic) * Patients must have measurable or non-measurable stage III/locally advanced or metastatic carcinoma of the breast where local therapy with curative intent is not possible; lesions must be evaluated =\< 4 weeks prior to study randomization; diagnostic-quality computed tomography (CT) scans with both oral and intravenous (IV) contrast are the expected radiologic method, unless an alternative is approved * NOTE: Where baseline imaging has already been performed =\< 6 weeks prior to study randomization, repeat imaging may not be required * NOTE: As of October 16, 2016, accrual of new patients having non-measurable disease has stopped; the planned accrual for this target population has been reached * Pre/peri- and postmenopausal women and all men are eligible for this trial; postmenopausal is defined as: * Age \>= 5…