We consider the problem of energy aware scheduling of a set of jobs on a set of unrelated parallel machines with the average weighted completion time plus energy objective. The processing time and the energy consumption of the jobs are machine and speed dependent. Also, every job is subject to a machine-dependent release date. Firstly, we aim to find a non-preemptive schedule of the jobs minimizing the average weighted completion time plus energy, and we propose a randomized approximation algorithm that we derandomize obtaining a deterministic approximation algorithm. We then consider the budget variant of the problem where the objective is to minimize the average completion time while the total energy consumption does not exceed a given budget.