This article investigates the performance of an ultra-dense network (UDN) from an energy-efficiency (EE) standpoint leveraging the interplay between stochastic geometry (SG) and mean-field game (MFG) theory. In this setting, base stations (BSs) (resp. users) are uniformly distributed over a two-dimensional plane as two independent homogeneous Poisson point processes (PPPs), where users associate to their nearest BSs. The goal of every BS is to maximize its own energy efficiency subject to channel uncertainty, random BS location, and interference levels. Due to the coupling in interference, the problem is solved in the mean-field (MF) regime where each BS interacts with the whole BS population via time-varying MF interference. As a main contribution, the asymptotic convergence of MF interference to zero is rigorously proved in a UDN with multiple transmit antennas. It allows us to derive a closed-form EE representation, yielding a tractable EE optimal power control policy. This proposed power control achieves more than 1.5 times higher EE compared to a fixed power baseline.