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Diversification rates vary over time, yet the factors driving these variations remain unclear. Temporal declines in speciation rates have often been interpreted as the effect of ecological limits, competition, and diversity dependence, emphasising the role of biotic factors. Abiotic factors, such as climate change, are also supposed to have affected diversification rates over geological time scales, yet direct tests of these presumed effects have mainly been limited to few clades well represented in the fossil record. If warmer climatic periods have sustained faster speciation, this could explain slowdowns in speciation during the Cenozoic climate cooling. Here, we apply state-of-the art diversity-dependent and temperature-dependent phylogenetic models of diversification to 218 tetrapod families, along with constant rate and time-dependent models. We confirm the prevalence of diversification slowdowns, and find as much support for temperature-dependent than diversity-dependent models. These results call for a better integration of these two processes in studies of diversification dynamics.