We design optimal privacy-enhancing and cost-efficient energy management strategies for consumers that are equipped with a rechargeable energy storage. The Kullback-Leibler divergence rate is used as privacy measure and the expected cost-saving rate is used as utility measure. The corresponding energy management strategy is designed by optimizing a weighted sum of both privacy and cost measures over a finite time horizon, which is achieved by formulating our problem into a belief-state Markov decision process problem. A computationally efficient approximated Q-learning method is proposed as a generalization to high-dimensional problems over an infinite time horizon. At last, we explicitly characterize a stationary policy that achieves the steady belief state over an infinite time horizon, which greatly simplifies the design of the privacy-preserving energy management strategy. The performance of the practical design approaches are finally illustrated in numerical experiments.