Entinostat, Lapatinib Ditosylate and Trastuzumab in Treating Patients With Locally Recurrent or Distant Relapsed Metastatic Breast Cancer Previously Treated With Trastuzumab Only

Inclusion Criteria: * Patients have histological confirmation of invasive breast carcinoma * Patients have locally recurrent or distant relapsed metastatic disease * Patients have positive HER2 expression by immunohistochemistry (IHC) (3+) or fluorescence in situ hybridization (FISH) testing (\> 2.0 ratio) * Patients are able to swallow and retain oral medication (i.e., no uncontrolled vomiting, inability to swallow, or diagnosis of chronic malabsorption) * Patients have Eastern Cooperative Oncology Group (ECOG) performance status 0-1 * Patients must have received prior trastuzumab for \> 2 month period before disease recurrence or recurrence or progression while on trastuzumab-based therapy * Patients have ability and willingness to sign written informed consent * Female patients of childbearing potential (a female not free from menses \> 2 years or not surgically sterilized) must be willing to use an adequate barrier method of contraception to prevent pregnancy or agree to abstain from heterosexual activity throughout the study; male patients who are able to father children must use an adequate barrier method of contraception * Female patients of childbearing potential must have negative serum pregnancy test within 14 days of starting protocol therapy * Patients with brain metastasis have no signs of progressive disease 4 months after the completion of brain metastasis treatment (radiation therapy, surgery, etc.) do not require anticonvulsants or corticosteroids, and have been off such drugs for at least 7 days * Both men and women and members of all races and ethnic groups are eligible for this trial Exclusion Criteria: * Patients are receiving concurrent anti-cancer therapy (chemotherapy, immunotherapy, biological therapy and hormonal therapy) while taking study medication * Serum bilirubin \>= 1.5 x upper limit of normal (ULN) * Alanine aminotransferase (ALT) and aspartate aminotransferase (AST) \>= 3 x ULN (with or without liver metastasis \[mets\]) * Absolute neutrophil count (ANC) \< 1.5 * Hemoglobin =\< 9 * Platelet =\< 140,000 * Patients have an active infection and require intravenous (IV) or oral antibiotics * Cardiac arrhythmia requiring maintenance medication * History of gastrointestinal disorders (medical disorders or extensive surgery) which may interfere with the absorption of the study drug * Patients have a concurrent disease or condition that would make them inappropriate for study participation, or any serious medical disorder that would interfere with patients' safety * Serum creatinine \> 2.0 mg/dL