This phase I trial studies the side effects and best dose of entinostat when given together with lapatinib ditosylate and trastuzumab in treating patients with breast cancer that has spread from the original (primary) tumor to distant organs or distant lymph nodes or has recurred (come back) at or near the same place as the original (primary) tumor, usually after a period of time during which the cancer could not be detected. Entinostat and lapatinib ditosylate may stop the growth of tumor cells by blocking some of the enzymes needed for cell growth. Monoclonal antibodies, such as trastuzumab, may interfere with the ability of tumor cells to grow and spread. Giving entinostat together with lapatinib ditosylate and trastuzumab may kill more tumor cells.

PRIMARY OBJECTIVES:

I. To determine the recommended phase II dose (RP2D) for entinostat in combination with lapatinib (lapatinib ditosylate) in patients whom trastuzumab has failed for human epidermal growth factor receptor 2+ (HER2+) metastatic breast cancer (Phase I).

II. To determine the maximum tolerated dose (MTD) for entinostat in combination with lapatinib and trastuzumab in patients whom trastuzumab has failed for HER2+ metastatic breast cancer (Phase I Trastuzumab Cohort).

SECONDARY OBJECTIVES:

I. To determine the toxicity of combination therapy with entinostat and lapatinib in patients whom trastuzumab has failed for HER2+ metastatic breast cancer (Phase I).

II. To determine the toxicity of entinostat in combination with lapatinib and trastuzumab in patients whom trastuzumab has failed for HER2+ metastatic breast cancer (Phase I Trastuzumab Cohort).

EXPLORATORY OBJECTIVES:

I. Determine whether the 2-drug combination modulates the expression of HER2, phosphorylated HER2 (pHER), epidermal growth factor receptor (EGFR), phosphorylated EGFR (pEGFR), v-akt murine thymoma viral oncogene homolog 1 (Akt), and phosphorylated Akt (pAkt) in breast tumors and/or circulating tumor cells (CTCs).

OUTLINE: This is a dose-escalation study of entinostat.

Patients receive entinostat orally (PO) on days 1 and 15 and lapatinib tosylate PO on days 1-28. Courses repeat every 28 days in the absence of disease progression or unacceptable toxicity. Patients in the Phase I trastuzumab cohort also receive maintenance dose of trastuzumab intravenously (IV) over 30-90 minutes every 3 weeks.

After completion of study treatment, patients are followed up for 28 days or until toxicities are resolved.

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