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With the rapid development of the Internet of Things (IoT), computational workloads are gradually moving toward the internet edge for low latency. Due to significant workload fluctuations, edge data centers built in distributed locations suffer from resource underutilization and requires capacity underprovisioning to avoid wasting capital investment. The workload fluctuations, however, also make edge data centers more suitable for battery-assisted power management to counter the performance impact due to underprovisioning. In particular, the workload fluctuations allow the battery to be frequently recharged and made available for temporary capacity boosts. But, using batteries can overload the data center cooling system which is designed with a matching capacity of the power system. In this paper, we design a novel power management solution, DeepPM, that exploits the UPS battery and cold air inside the edge data center as energy storage to boost the performance. DeepPM uses deep reinforcement learning (DRL) to learn the data center thermal behavior online in a model-free manner and uses it on-the-fly to determine power allocation for optimum latency performance without overheating the data center. Our evaluation shows that DeepPM can improve latency performance by more than 50% compared to a power capping baseline while the server inlet temperature remains within safe operating limits (e.g., 32°C).