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Wireless networking using GHz or THz spectra has encouraged mobile service providers to deploy small cells to improve link quality and cell capacity using mmWave backhaul links. As green networking for less CO2 emission is mandatory to confront global climate change, we need energy efficient network management for such denser small-cell heterogeneous networks (HetNets) that already suffer from observable power consumption. We establish a dual-objective optimization model that minimizes energy consumption by switching off unused small cells while maximizing user throughput, which is a mixed integer linear problem (MILP). Recently, the deep reinforcement learning (DRL) algorithm has been applied to many NP-hard problems of the wireless networking field, such as radio resource allocation, association and power saving, which can induce a near-optimal solution with fast inference time as an online solution. In this paper, we investigate the feasibility of the DRL algorithm for a dual-objective problem, energy efficient routing and throughput maximization, which has not been explored before. We propose a proximal policy (PPO)-based multi-objective algorithm using the actor-critic model that is realized as an optimistic linear support framework in which the PPO algorithm searches for feasible solutions iteratively. Experimental results show that our algorithm can achieve throughput and energy savings comparable to the CPLEX. [ABSTRACT FROM AUTHOR]