We consider updating strategies for a local cache which downloads time-sensitive files from a remote server through a bandwidth-constrained link. The files are requested randomly from the cache by local users according to a popularity distribution which varies over time according to a Markov chain structure. We measure the freshness of the requested time-sensitive files through their Age of Information (AoI). The goal is then to minimize the average AoI of all requested files by appropriately designing the local cache's downloading strategy. To achieve this goal, the original problem is relaxed and cast into a Constrained Markov Decision Problem (CMDP), which we solve using a Lagrangian approach and Linear Programming. Inspired by this solution for the relaxed problem, we propose a practical cache updating strategy that meets all the constraints of the original problem. Under certain assumptions, the practical updating strategy is shown to be optimal for the original problem in the asymptotic regime of a large number of files. For a finite number of files, we show the gain of our practical updating strategy over the traditional square-root-law strategy (which is optimal for fixed non time-varying file popularities) through numerical simulations.