Fast convergence routing is a critical issue for Low Earth Orbit (LEO) constellation networks because these networks have dynamic topology changes, and transmission requirements can vary over time. However, most of the previous research has focused on the Open Shortest Path First (OSPF) routing algorithm, which is not well-suited to handle the frequent changes in the link state of the LEO satellite network. In this regard, we propose a Fast-Convergence Reinforcement Learning Satellite Routing Algorithm (FRL–SR) for LEO satellite networks, where the satellite can quickly obtain the network link status and adjust its routing strategy accordingly. In FRL–SR, each satellite node is considered an agent, and the agent selects the appropriate port for packet forwarding based on its routing policy. Whenever the satellite network state changes, the agent sends “hello” packets to the neighboring nodes to update their routing policy. Compared to traditional reinforcement learning algorithms, FRL–SR can perceive network information faster and converge faster. Additionally, FRL–SR can mask the dynamics of the satellite network topology and adaptively adjust the forwarding strategy based on the link state. The experimental results demonstrate that the proposed FRL–SR algorithm outperforms the Dijkstra algorithm in the performance of average delay, packet arriving ratio, and network load balance.