Quantifying the equilibrium response of global temperatures to an increase in atmospheric carbon dioxide concentrations is one of the cornerstones of climate research. Components of the Earth's climate system that vary over long timescales, such as ice sheets and vegetation, could have an important effect on this temperature sensitivity, but have often been neglected. Here we use a coupled atmosphere-ocean general circulation model to simulate the climate of the mid-Pliocene warm period (about three million years ago), and analyse the forcings and feedbacks that contributed to the relatively warm temperatures. Furthermore, we compare our simulation with proxy records of mid-Pliocene sea surface temperature. Taking these lines of evidence together, we estimate that the response of the Earth system to elevated atmospheric carbon dioxide concentrations is 30-50% greater than the response based on those fast-adjusting components of the climate system that are used traditionally to estimate climate sensitivity. We conclude that targets for the long-term stabilization of atmospheric greenhouse-gas concentrations aimed at preventing a dangerous human interference with the climate system should take into account this higher sensitivity of the Earth system.