The sensitivity of tropical forest carbon to climate is a key uncertainty in predicting global climate change. While short-term drying and warming are known to impact forests it is unknown if such effects translate into long-term responses. Here we analyse 590 permanent plots measured across the tropics to derive the equilibrium climate controls on forest carbon. Maximum temperature is the most important predictor of aboveground biomass (-9.1 Mg C ha-1 °C-1), primarily by reducing woody productivity, and with a greater rate of decline in the hottest forests (>32.2 °C). Our results nevertheless reveal greater thermal resilience than observations of short-term variation imply. To realise the long-term climate adaptation potential of tropical forests requires both protecting them and stabilising the Earth’s climate.