This paper addresses the problem of scheduling different non-preemptive jobs on a single machine under time of use electricity tariffs consideration. The considered machine has three main states (OFF, ON, Idle) and two transition states (Turn-on and Turn-off). Each of these machine’s states as well as the processing jobs, consume a specific amount of energy. Moreover, a speed-scalable case of the problem is considered, in which jobs can be processed at an arbitrary speed with a trade-off between speed and energy consumption. First, a mixed-integer linear programming model with the objective of minimizing total energy consumption costs formulates this scheduling problem. Then, since this problem is strongly NP-hard, different approximate optimization methods are investigated to provide near-optimal solutions. Finally, an extensive computational study is carried out to establish the efficiency of the proposed algorithms. The obtained results show, how a well-tuned genetic algorithm combined with an adequate local search procedure constitutes an efficient method for solving this energy-aware scheduling problem.