The development of mega constellations inevitably brings various problems for the development of routing techniques. Most of the existing work considers end-to-end delay and load balancing problems, while the analysis of routing strategies in case of link performance degradation is neglected, and an optimization approach applicable to mega satellite networks is not developed. In this letter, we propose a robust routing strategy based on deep reinforcement learning (RRS-DRL) that regards the Age of Information (AoI) of packets as an optimization target, and ensures the effectiveness of message transmission throughout the network. Extensive simulation results show that our proposed RRS-DRL algorithm obtains a lower average AoI across the network and better utilization of the resources than the traditional shortest path algorithm, significantly increasing the robustness of the constellation.