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In this paper, we propose a spatial-temporal learning-based distributed routing framework for dynamic Low Earth Orbit (LEO) satellite networks, where graph attention networks (GAT) and long short-term memory (LSTM) are integrated within a deep Q-network (DQN)-based architecture to enable distributed and adaptive routing decisions based on local observations. The routing problem is formulated as a partially observable Markov decision process (POMDP) to address partial observability under dynamic topology and time-varying traffic. Simulation results show that the proposed method significantly outperforms conventional and learning-based routing schemes in terms of throughput, packet loss, queue length, and end-to-end delay, while achieving proactive congestion avoidance with up to 23.26% queue reduction. In addition, the proposed approach maintains low computational overhead with negligible carbon emissions, demonstrating its efficiency from a Green AI perspective.