Energy-aware distributed heterogeneous flexible job shop scheduling (DHFJS) problem is an extension of the traditional FJS, which is harder to solve. This work aims to minimize total energy consumption (TEC) and makespan for DHFJS. A deep $Q$ -networks-based co-evolution algorithm (DQCE) is proposed to solve this NP-hard problem, which includes four parts: First, a new co-evolutionary framework is proposed, which allocates sufficient computation to global searching and executes local search surrounding elite solutions. Next, nine problem features-based local search operators are designed to accelerate convergence. Moreover, deep $Q$ -networks are applied to learn and select the best operator for each solution. Furthermore, an efficient heuristic method is proposed to reduce TEC. Finally, 20 instances and a real-world case are employed to evaluate the effectiveness of DQCE. Experimental results indicate that DQCE outperforms the six state-of-the-art algorithms for DHFJS.