Satellite caching integrates Low Earth Orbit (LEO) satellite communication with edge computing to enable efficient content delivery by caching content on LEO satellites, particularly benefiting users without access to ground infrastructure. However, the rapid movement of LEO satellites leads to highly dynamic network topologies. This naturally poses significant challenges, making traditional ground edge caching strategies ineffective in such environments. To address this issue, we focus on the problem of Network Topology-aware Dynamic Satellite Caching (NTDSC) in LEO satellite networks. By leveraging the temporal network model to capture the time-varying characteristics of satellite connectivity, we formulate the NTDSC problem as a constrained optimization problem that accounts for topological stability and centrality metrics. Our goal is to minimize the average transmission delay and average transmission energy consumption across satellite links while meeting constraints on the storage capacity of each LEO satellite and the maximum number of replicas allowed for each content. Then, we propose LEOCaching, a Multi-Agent Reinforcement Learning (MARL) based approach for effectively solving the NTDSC problem. LEOCaching exploits deep reinforcement learning with spatial and temporal feature extraction to enable adaptive and topology-aware satellite caching decisions in dynamic satellite networks. Finally, experimental results demonstrate that LEOCaching achieves lower average transmission delay and average transmission energy consumption compared to existing approaches.