Deploying a radio frequency (RF) access point (AP) to the visible light communication (VLC) system is a promising strategy to overcome the VLC's limitations, such as limited coverage, strictly line-of-sight transmission, and mobility robustness, etc. In this paper, we focus on the energy-aware design of network selection and resource allocation for a heterogeneous network combining with RF and VLC APs. For adapting to different timescale network states and stochastic data arrival, we propose an on-line two-timescale adaptive network resource optimization (ANRO) framework by employing the Lyapunov optimization technique. At the large timescale, we first develop a closed-form solution for the subproblem of network selection for user equipment. Second, we design a cost-effective and easy-to-realize algorithm for VLC's joint transmission scheduling and power control subproblem, which is a nonconvex optimization. While at the small timescale, we obtain the optimal solution for RF's joint resource block and power allocation subproblem, which is proven a mixed integer nonlinear optimization. Simulation results demonstrate that the ANRO can achieve a tradeoff between network power consumption and delay. Furthermore, it not only can stabilize the network but also can significantly reduce the energy consumption compared with other existing schemes.