Atmospheric carbon dioxide concentrations seem to have been several times modern levels during much of the Palaeozoic era (543-248million years ago), but decreased during the Carboniferous period to concentrations similar to that of today. Given that carbon dioxide is a greenhouse gas, it has been proposed that surface temperatures were significantly higher during the earlier portions of the Palaeozoic era. A reconstruction of tropical sea surface temperatures based on the δ18O of carbonate fossils indicates, however, that the magnitude of temperature variability throughout this period was small, suggesting that global climate may be independent of variations in atmospheric carbon dioxide concentration. Here we present estimates of sea surface temperatures that were obtained from fossil brachiopod and mollusc shells using the `carbonate clumped isotope' method-an approach that, unlike the δ18O method, does not require independent estimates of the isotopic composition of the Palaeozoic ocean. Our results indicate that tropical sea surface temperatures were significantly higher than today during the Early Silurian period (443-423Myr ago), when carbon dioxide concentrations are thought to have been relatively high, and were broadly similar to today during the Late Carboniferous period (314-300Myr ago), when carbon dioxide concentrations are thought to have been similar to the present-day value. Our results are consistent with the proposal that increased atmospheric carbon dioxide concentrations drive or amplify increased global temperatures.