This paper focuses on the development of energy efficient visible light communication (VLC) networks. More specifically, since the quality-of-service and energy cost are key parameters in designing energy efficient networks, this paper optimizes the VLC base station (BS) density to minimize the area power consumption (APC) under an outage probability constraint. Using stochastic geometry, approximations of the outage probability of VLC networks are first introduced. The derived approximations are applicable to an arbitrary field-of-view at photodiodes and present low computational complexities. Leveraging the derived analytical results, a low complexity algorithm to find the VLC BS density that minimizes the APC of VLC networks is then proposed. The numerical simulations corroborate the tightness of the approximations on the outage probability and confirm that the proposed algorithm exhibits almost identical performance as the algorithm that exhaustively searches the optimal BS density.