With the soaring growth of data traffic, unmanned aerial vehicle (UAV) based edge caching has been regarded as a promising solution to alleviate network congestion and enable users to obtain their desired contents with reduced delay. For the UAV-based edge caching, how to jointly plan the trajectory and caching strategy is the key, which determines how much benefit can achieve accordingly. Such a joint strategy design highly depends on the content demands in the network. However, the content demands are usually heterogeneous both temporally and spatially, and hardly known in advance. Such demand uncertainty makes the joint strategy design extremely challenging. In this paper, aiming at maximizing the reduced delay brought by the UAV-based edge caching, we propose a proactive joint trajectory and caching strategy under uncertain content demands. We formulate it into a risk-averse stochastic optimization problem to guarantee the maximal benefit with a high probability. Furthermore, considering the fact that the precise distributional information might be unavailable in practice, we focus on the worst case and develop a data-driven distributionally robust solution, making the strategy trustworthy. Simulation results demonstrate the effectiveness of the proposed strategy.