The unmanned aerial vehicles (UAVs) are able to assist damaged cellular networks as aerial base stations (ABSs) due to their flexibility and affordability. This paper investigates the cache-enabled UAV emergency communication networks in the post-disaster area using stochastic geometry approaches. We propose a cache-enabled UAV emergency communication network framework in which a limited number of ABSs are deployed overlaid the partially destructed cellular network. The ABSs are equipped with cache units that adopt probabilistic caching strategy. We design a content-centric association strategy within the specified circular cache area by comprehensively considering the cache status and channel conditions. Based on the framework, we derive the exact expressions of the offloading probability and average rate under the general channel model taking line-of-sight (LOS)/ non-line-of-sight (NLoS) path loss and Nakagami-m fading into account. Simulation results demonstrate that there exist optimal values for ABSs number, ABSs altitude, and cache serving radius to maximize the offloading probability and average rate. Moreover, increasing ABSs number and the cache serving radius in the case that ABSs are deployed at the relatively low altitude is beneficial to improve the system performance. A large number of ABSs helps to reduce the optimal cache serving radius for maximizing the system performance.