The proliferation of real-time applications has motivated extensive research on analyzing and optimizing data freshness in the context of age of information. However, classical frameworks of privacy (e.g., differential privacy (DP)) have overlooked the impact of data freshness on privacy guarantees, and hence may lead to unnecessary accuracy loss when trying to achieve meaningful privacy guarantees in time-varying databases. In this work, we introduce age-dependent DP, taking into account the underlying stochastic nature of a time-varying database. In this new framework, we establish a connection between classical DP and age-dependent DP, based on which we characterize the impact of data staleness and temporal correlation on privacy guarantees. Our characterization demonstrates that aging, i.e., using stale data inputs and/or postponing the release of outputs, can be a new strategy to protect data privacy in addition to noise injection in the traditional DP framework. Furthermore, to generalize our results to a multi-query scenario, we present a sequential composition result for age-dependent DP. We then characterize and achieve the optimal tradeoffs between privacy risk and utility. Finally, case studies show that, when achieving a target of an arbitrarily small privacy risk in a single-query case, the approach of combining aging and noise injection can achieve a bounded accuracy loss, whereas using noise injection only (as in the DP benchmark) will lead to an unbounded accuracy loss.