With the explosive growth of low earth orbit (LEO) satellite networks, such as Starlink, satellite communication has lower latency and can achieve high-speed transmission than before. However, the time-variant topology during all network lifetimes makes the routing problem in the LEO satellite networks challenging. Therefore, in this paper, we propose the deep reinforcement learning-based satellite routing (DRL-SR) method to tackle the multi-commodity flow routing problem in the LEO satellite networks. Given the current state of the satellite network environment, the satellite operation center will determine how to route the requests to the matching destinations. Particularly, the single agent in our DRL-SR approach can determine the multiple next hops as actions for all the corresponding requests each timeslot. Finally, simulation results show that our proposed algorithm yields lower latency than the shortest path approach.