Smart meters (SMs) play a pivotal rule in the smart grid by being able to report the electricity usage of consumers to the utility provider (UP) almost in real-time. However, this could leak sensitive information about the consumers to the UP or a third-party. Recent works have leveraged the availability of energy storage devices, e.g., a rechargeable battery (RB), in order to provide privacy to the consumers with minimal additional energy cost. In this paper, a privacy-cost management unit (PCMU) is proposed based on a model-free deep reinforcement learning algorithm, called deep double Q-learning (DDQL). Empirical results evaluated on actual SMs data are presented to compare DDQL with the state-of-the-art, i.e., classical Q-learning (CQL). Additionally, the performance of the method is investigated for two concrete cases where attackers aim to infer the actual demand load and the occupancy status of dwellings. Finally, an abstract information-theoretic characterization is provided.