Targeting Aurora A Kinase to Overcome Treatment Resistance in Advanced HR+/HER2+ Breast Cancer

The goal of this clinical trial is to learn if alisertib in addition to usual care works to treat HR+/HER2+ breast cancer. The main questions it aims to answer are: * Does alisertib stop the communication between HR and HER2? * Are there genetic markers that predict how well someone's cancer will respond to alisertib? Participants will receive alisertib in addition to their usual care.

Inclusion Criteria: * Age \> 18 years at the time of consent. * ECOG performance status \/=60%). * Metastatic HR+/HER2+ breast cancer (estrogen receptor (ER) and/or progesterone receptor (PR) ≥1%, HER2 3+ by immunohistochemistry (IHC) or amplified by in situ hybridization (ISH). * Prior standard induction treatment with chemotherapy + trastuzumab + pertuzumab (HP) or fam-trastuzumab deruxtecan (T-DXd) and have completed a minimum of 4 cycles without progressive disease. * Planned to start or are receiving endocrine therapy + HER2-directed (HP or pertuzumab/trastuzumab/hyaluronidase-zzxf (PHESGO®)) therapy. * Demonstrate adequate organ function; all screening labs to be obtained within 28 days prior to registration. * Left ventricular ejection fraction ≥ 50% as assessed by echocardiogram (ECHO) or multi-gated acquisition scan (MUGA) documented within 6 weeks prior to the study treatment. * Patients must have measurable disease by Response Evaluation Criteria in Solid Tumors volume 1.1 (RECIST 1.1) or evaluable disease with circulating tumor DNA (ctDNA) that is detectible. Measurable disease is defined as at least one lesion that can be accurately measured in at least one dimension (longest diameter to be recorded for non-nodal lesions and short axis for nodal lesions) as ≥20 mm (≥2 cm) by chest x-ray or as ≥10 mm (≥1 cm) with CT scan, MRI, or calipers by clinical exam. * Willingness to receive growth factor injections for neutropenia prophylaxis. Only patients without any grade 3 or higher neutropenia during induction chemotherapy or T-DXd will be permitted to proceed without prophylactic growth factor support. If the enrolled patients in this trial develop high grade or prolonged neutropenia, the addition of growth factor will be required. Exclusion Criteria: * Active infection requiring systemic therapy. * Patients with a prior or concurrent malignancy whose natural history or treatment has the potential to interfere with the safety or efficacy assessment of the investigational regimen are not eligible for this trial. * Treatment with any investigational drug within 14 days prior to registration, or within 5 half-lives of the investigational product, whichever is longer.