The deployment of deep learning (DL) models for precoding in massive multiple-input multiple-output (mMIMO) systems is often constrained by high computational demands and energy consumption. In this paper, we investigate the compute energy efficiency of mMIMO precoders using DL-based approaches, comparing them to conventional methods such as zero forcing and weighted minimum mean square error (WMMSE). Our energy consumption model accounts for both memory access and calculation energy within DL accelerators. We propose a framework that incorporates mixed-precision quantization-aware training and neural architecture search to reduce energy usage without compromising accuracy. Using a ray-tracing dataset covering various base station sites, we analyze how site-specific conditions affect the energy efficiency of compressed models. Our results show that deep neural network compression generates precoders with up to 35 times higher energy efficiency than WMMSE at equal performance, depending on the scenario and the desired rate. These results establish a foundation and a benchmark for the development of energy-efficient DL-based mMIMO precoders.