In low earth orbit (LEO) satellite-based applications (e.g., remote sensing and surveillance), it is important to efficiently transmit collected data to ground stations (GS). However, LEO satellites’ high mobility and resultant insufficient time for downloading make this challenging. In this paper, we propose a deep-reinforcement-learning (DRL)-based cooperative downloading scheme, which utilizes inter-satellite communication links (ISLs) to fully utilize satellites’ downloading capabilities. To this end, we formulate a Markov decision problem (MDP) with the objective to maximize the amount of downloaded data. To learn the optimal approach to the formulated problem, we adopt a soft-actor-critic (SAC)-based DRL algorithm in discretized action spaces. Moreover, we design a novel neural network consisting of a graph attention network (GAT) layer to extract latent features from the satellite network and parallel fully connected (FC) layers to control individual satellites of the network. Evaluation results demonstrate that the proposed DRL-based cooperative downloading scheme can enhance the average utilization of contact time by up to 17.8% compared with independent downloading and randomly offloading schemes.