In-networking caching, as one of the critical features of the Information-Centric Networking (ICN), has been widely investigated. On-path caching is a light-weight implementation of in-networking caching. Existing studies on on-path caching strategy are mainly confined to the local information and neglect the interaction between multi-node cache decisions, which incurs cache redundancy and lead to performance degradation. To this end, we analyze the relationship between multi-node decisions and propose a lightweight on-path cooperative caching strategy, called GAC. In GAC, caching decisions are divided into two phases. In the first round, the upstream node makes caching decision based on the downstream caching information. In the second round, according to the feedback from upstream nodes, each node is given an opportunity to optimize the caching decision it made before. We evaluate the performance of our scheme through extensive simulations regarding a wide range of performance metrics. The experimental results indicate GAC can achieve significant performance improvement compared with representative approaches in terms of server load reduction ratio, average hop reduction ratio and average cache hit ratio. Meanwhile, GAC can significantly reduce the number of cache evictions.