Taking up the ongoing shift towards green production, this article addresses energy-oriented flexible job shop scheduling. Existing approaches mainly focus on single objectives in terms of energy utilization such as minimizing energy consumption. However, production control can affect multiple energy-related criteria. Therefore, we propose a flexible job shop scheduling model to minimize real-time pricing-related energy costs, peak demand and energy-related emissions. Motivated by the reported preeminence of Constraint Programming (CP) for a variety of scheduling problems, we extend a CP formulation for our study. To evaluate potential contradictory between energy objectives, we present nine objective functions by means of different lexicographic orders. In addition, we enhance the proposed scheduling model to account for sequence-dependent setup and due dates. To analyze and compare the effectiveness of the different model formulations, we present computational experiments for 20 small-, medium- and large-sized problem instances. Our study indicates that productivity can be maximized while, on average, energy costs are reduced by 5.3%, peak demand by 11.8%, emissions by 8.3% compared to traditional job scheduling. However, partly conflicting objectives require the decision maker to select the objective function most suitable to the individual needs. Including setup effort and due date compliance into energy-aware scheduling is possible and needed to make the concept of energy-aware scheduling applicable to industrial practice. We show that the additional aspects limit the potential improvement. Hence, it is crucial to understand such complex scheduling systems combining energy awareness, setup and due date compliance.