This randomized phase III trial studies exemestane and entinostat to see how well they work compared to exemestane alone in treating patients with hormone receptor-positive breast cancer that has spread to nearby tissue or lymph nodes (locally advanced) or another place in the body (metastatic). Estrogen can cause the growth of breast cancer cells. Endocrine therapy using exemestane may fight breast cancer by lowering the amount of estrogen the body makes. Entinostat may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. It is not yet known whether exemestane is more effective with or without entinostat in treating breast cancer.