Abemaciclib in Treating Patients With Advanced, Refractory, and Unresectable Digestive System Neuroendocrine Tumors

Inclusion Criteria: * Histologically confirmed GEP NET, radiographically progressed on at least one line of standard therapy within the past 12 months * Primary tumors may be in: pancreas, foregut (esophagus, stomach, duodenum), midgut (small intestine, appendix), hindgut (large intestine, rectum), or unknown origin * Tumors may be functional (associated with clinical symptoms of hormone secretion) or non-functional * Well-differentiated NET with low grade (Ki67 index \< 3% or mitotic index \< 2 mitoses/10 high power field \[HPF\]), intermediate grade (Ki67 index 3-20% or mitotic index 2-20 mitoses/10 HPF), or high grade (Ki67 21% to ≤ 55% of mitotic index 21-55% mitoses/10 HPF). In cases where pathology reports call out only a "high grade neuroendocrine carcinoma", such patients are eligible only if well differentiated status is confirmed by a board-certified pathologist AND Ki-67 is ≤ 55% * Metastatic or locally advanced unresectable disease * Measurable disease by computed tomography (CT) or magnetic resonance imaging (MRI) as per Response Evaluation Criteria in Solid Tumors (RECIST) version (v) 1.1 * Prior or concurrent therapy with somatostatin analogs (SSAs) is allowed. If concurrent therapy, dose must be stable for at least 2 months * Patients with carcinoid syndrome must have symptoms controlled with stable doses of SSAs for at least 2 months \* Telotristat is not allowed * Age \>= 18 years * Eastern Cooperative Oncology Group (ECOG) performance status of 0 t…