As environmental awareness grows, energy-aware scheduling is attracting increasing attention. This paper investigates the flexible job shop scheduling problem with sequence-dependent setup times and transportation times (FJSP-SDST-T) and the objective is to minimize total energy consumption. To begin with, the total energy consumption of the workshop is analyzed and a novel mixed integer linear programming (MILP) model is formulated. Due to that FJSP-SDST-T is NP-hard, an effective hybrid algorithm (HGA) that hybridizes the genetic algorithm (GA) and variable neighborhood search (VNS) algorithm is proposed to solve the problem specifically for that with large size. HGA takes advantage of the good global searching ability of GA and the powerful local searching ability of VNS, and it can have a good balance of intensification and diversification. Then, four energy-conscious decoding methods are designed, in which two energy-saving strategies namely postponing strategy and Turn Off/On strategy are specially designed according to the characteristics of FJSP-SDST-T. Finally, experiments are carried out and the results show the effectiveness of the MILP model, the energy-conscious decoding methods and HGA.