Hyperbolic neural networks (HNNs) have demonstrated notable efficacy in representing real-world data with hierarchical structures via exploiting the geometric properties of hyperbolic spaces characterized by negative curvatures. Curvature plays a crucial role in optimizing HNNs. Inappropriate curvatures may cause HNNs to converge to suboptimal parameters, degrading overall performance. So far, the theoretical foundation of the effect of curvatures on HNNs has not been developed. In this paper, we derive a PAC-Bayesian generalization bound of HNNs, highlighting the role of curvatures in the generalization of HNNs via their effect on the smoothness of the loss landscape. Driven by the derived bound, we propose a sharpness-aware curvature learning method to smooth the loss landscape, thereby improving the generalization of HNNs. In our method,