We study the problem of proactively pushing contents into a finite capacity cache memory of a user equipment in order to reduce the long-term average energy consumption in a wireless network. We consider an online social network (OSN) framework, in which new contents are generated over time and each content remains relevant to the user for a random time period, called the lifetime of the content. The user accesses the OSN through a wireless network at random time instants to download and consume all the relevant contents. Downloading contents has an energy cost that depends on the channel state and the number of downloaded contents. Our aim is to reduce the long-term average energy consumption by proactively caching contents at favorable channel conditions. In previous work, it was shown that the optimal caching policy is infeasible to compute (even with the complete knowledge of a stochastic model describing the system), and a simple family of threshold policies was introduced and optimised using the finite difference method. In this paper we improve upon both components of this approach: we use linear function approximation (LFA) to better approximate the considered family of caching policies, and apply the REINFORCE algorithm to optimise its parameters. Numerical simulations show that the new approach provides reduction in both the average energy cost and the running time for policy optimisation.