This paper introduces a new efficient autopre-coder (AP) based deep learning approach for massive multiple-input multiple-output (mMIMO) downlink systems in which the base station is equipped with a large number of antennas with energy-efficient power amplifiers (PAs) and serves multiple user terminals. We present AP-mMIMO, a new method that jointly eliminates the multi-user interference and compensates the severe nonlinear (NL) PA distortions. Unlike previous works, AP-mMIMO has a low computational complexity, making it suitable for a global energy-efficient system. Specifically, we aim to design the PA-aware precoder and the receive decoder by leveraging the concept of autoprecoder, whereas the end-to-end massive multi-user (MU)-MIMO downlink is designed using a deep neural network (NN). Most importantly, the proposed AP-mMIMO is suited for the varying block fading channel scenario. To deal with such scenarios, we consider a two-stage precoding scheme: 1) a NN-precoder is used to address the PA non-linearities and 2) a linear precoder is used to suppress the multi-user interference. The NN-precoder and the receive decoder are trained off-line and when the channel varies, only the linear precoder changes on-line. This latter is designed by using the widely used zero-forcing precoding scheme or its low-complexity version based on matrix polynomials. Numerical simulations show that the proposed AP-mMIMO approach achieves competitive performance with a significantly lower complexity compared to existing literature.