Generally, epilepsy is considered as abnormally enhanced neuronal excitability and synchronization. So far, previous studies on the synchronization of epileptic brain networks mainly focused on the synchronization strength, but the synchronization stability has not yet been explored as deserved. In this paper, we propose a novel idea to construct a hypergraph brain network (HGBN) based on phase synchronization. Furthermore, we apply the synchronization stability framework of the nonlinear coupled oscillation dynamic model (generalized Kuramoto model) to investigate the HGBNs of epilepsy patients. Specifically, the synchronization stability of the epileptic brain is quantified by calculating the eigenvalue spectrum of the higher-order Laplacian matrix in HGBN. Results show that synchronization stability decreased slightly in the early stages of seizure but increased significantly prior to seizure termination. This indicates that an emergency self-regulation mechanism of the brain may facilitate the termination of seizures. Moreover, the variation in synchronization stability during epileptic seizures may be induced by the topological changes of epileptogenic zones (EZs) in HGBN. Finally, we verify that the higher-order interactions improve the synchronization stability of HGBN. This study proves the validity of the synchronization stability framework with the nonlinear coupled oscillation dynamical model in HGBN, emphasizing the importance of higher-order interactions and the influence of EZs on the termination of epileptic seizures.