Network robustness and throughput can be improved by routing each source-to-terminal (s, t) demand via two link-disjoint paths (TLDP). However, the use of TLDP incurs higher energy cost. Henceforth, we address the problem of minimizing the energy usage of networks that use TLDP. Specifically, our problem is to maximally switch off redundant network links while maintaining at least 0zT100% of (s, t) TLDP in the network, for a given T, and limiting the maximum link utilization (MLU) to no greater than a configured threshold. To address this problem, we present a fast heuristic, called TLDP by Shortest Path First (TLDP-SPF), and extensively evaluate its performance on both real and/or synthetic topologies and traffic demands. Our simulation results show that TLDP-SPF can reduce network energy usage, on average, by more than 20percent, even for MLU below 50 percent. As compared to using Shortest Path routing, while reducing energy by about 20percent, TLDP-SPF does not significantly affect (s, t) path length, even for MLU and lt 50 percent.