Biodiversity varies predictably with environmental energy around the globe, but the underlaying mechanisms remain incompletely understood. The evolutionary speed hypothesis predicts that environmental kinetic energy shapes variation in speciation rates through temperature- or life history-dependent rates of evolution. To test whether variation in evolutionary speed can explain the relationship between energy and biodiversity in birds, mammals, amphibians, and reptiles, we simulated diversification over 65 myr of geological and climatic change with a spatially explicit eco-evolutionary simulation model. We modeled four distinct evolutionary scenarios in which speciation-completion rates were dependent on temperature (M1), life history (M2), temperature and life history (M3), or were independent of temperature and life-history (M0). To assess the agreement between simulated and empirical data, we performed model selection by fitting supervised machine learning models to multidimensional biodiversity patterns. We show that a model with temperature-dependent rates of speciation (M1) consistently had the strongest support. In contrast to statistical inferences, which showed no general relationships between temperature and speciation rates in tetrapods, we demonstrate how process-based modeling can disentangle the causes behind empirical biodiversity patterns. Our study highlights how environmental energy has played a fundamental role in the evolution of biodiversity over deep time. [Biogeography; diversification; machine learning; macroevolution; molecular evolution; simulation.]