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The rapid development and expansion of the Internet of Things (IoT) paradigm has drastically increased the collection and exchange of data between sensors and systems, a phenomenon that raises serious privacy concerns in some domains. In particular, Smart Meters (SMs) share fine-grained electricity consumption of households with utility providers that can potentially violate users' privacy as sensitive information is leaked through the data. In order to enhance privacy, the electricity consumers can exploit the availability of physical resources such as a rechargeable battery (RB) to shape their power demand as dictated by a Privacy-Cost Management Unit (PCMU). In this paper, we present a novel method to learn the PCMU policy using Deep Reinforcement Learning (DRL). We adopt the mutual information (MI) between the user's demand load and the masked load seen by the power grid as a reliable and general privacy measure. Unlike previous studies, we model the whole temporal correlation in the data to learn the MI in its general form and use a neural network to estimate the MI-based reward signal to guide the PCMU learning process. This approach is combined with a model-free DRL algorithm known as the Deep Double Q-Learning (DDQL) method. The performance of the complete DDQL-MI algorithm is assessed empirically using an actual SMs dataset and compared with simpler privacy measures. Our results show significant improvements over state-of-the-art privacy-aware demand shaping methods.