  We consider a system consisting of a library of time-varying files, a server
that at all times observes the current version of all files, and a cache that
at the beginning stores the current versions of all files but afterwards has to
update %fresh versions of these files from the server. Unlike previous works,
the update duration is not constant but depends on the file and its Age of
Information (AoI), i.e., of the time elapsed since it was last updated. The
goal of this work is to design an update policy that minimizes the average AoI
of all files with respect to a given popularity distribution. Actually a
relaxed problem, close to the original optimization problem, is solved and a
practical update policy is derived. The update policy relies on the file
popularity and on the functions that characterize the update durations of the
files depending on their AoI. Numerical simulations show a significant
improvement of this new update policy compared to the so-called square-root
policy that is optimal under file-independent and constant update durations.
