Atmospheric carbon dioxide concentrations are believed to drive climate changes from glacial to interglacial modes, although geological and astronomical mechanisms have been invoked as ultimate causes. Additionally, it is unclear whether the changes between cold and warm modes should be regarded as a global phenomenon, affecting tropical and high-latitude temperatures alike, or if they are better described as an expansion and contraction of the latitudinal climate zones, keeping equatorial temperatures approximately constant. Here we present a reconstruction of tropical sea surface temperatures throughout the Phanerozoic eon (the past approximately 550 Myr) from our database of oxygen isotopes in calcite and aragonite shells. The data indicate large oscillations of tropical sea surface temperatures in phase with the cold-warm cycles, thus favouring the idea of climate variability as a global phenomenon. But our data conflict with a temperature reconstruction using an energy balance model that is forced by reconstructed atmospheric carbon dioxide concentrations. The results can be reconciled if atmospheric carbon dioxide concentrations were not the principal driver of climate variability on geological timescales for at least one-third of the Phanerozoic eon, or if the reconstructed carbon dioxide concentrations are not reliable.