Most of the world's cloud data service workloads are currently being backed by replicated state machines. Production-grade log replication protocols used for the job impose heavy data transfer duties on the primary server which need to disseminate the log commands to all the replica servers. UniCache proposes a principal solution to this problem using a learned replicated cache which enables commands to be sent over the network as compressed encodings. UniCache takes advantage of that each replica has access to a consistent prefix of the replicated log which allows them to build a uniform lookup cache used for compressing and decompressing commands consistently. UniCache achieves effective speedups, lowering the primary load in application workloads with a skewed data distribution. Our experimental studies showcase a low pre-processing overhead and the highest performance gains in cross-data center deployments over wide area networks.