Mobile edge computing (MEC) can use wireless access network (RAN) to provide users with nearby information technology (IT) services and cloud computing functions, which creates a high-performance and low latency service environment. By caching the popular content at small base station (SBS) can reduce the heavy backhaul load and the content retransmission. However, the time-varying and dynamic of the content requests may lead to the base station to cache the useless contents. In this paper, we study a distributed caching optimization problem in edge networks (ENs) with the spatio-temporal requirements. In the considered ENs, the cache control is described as a stochastic differential game (SDG) in which each SBS defines a caching strategy to reduce the total cost in terms of the service delay and backhaul link load. To reduce the computational complexity, the original optimization problem is transformed into a mean field game (MFG). We propose a distributed caching iterative control algorithm that decouples the information interaction between the general SBS and others through the mean field distribution. In addition, we obtain the optimal edge caching control strategy, while the existence and uniqueness of the mean field equilibrium (MFE) can also be guaranteed. Simulation results demonstrate that our proposed caching control algorithm can average reduce 27.12% storage cost and achieve better performance than other existing schemes.