Variations in global ice volume and temperature over the Cenozoic era have been investigated with a set of one-dimensional (1-D) ice-sheet models. Simulations include three ice sheets representing glaciation in the Northern Hemisphere, i.e. in Eurasia, North America and Greenland, and two separate ice sheets for Antarctic glaciation. The continental mean Northern Hemisphere surface-air temperature has been derived through an inverse procedure from observed benthic δ18O records. These data have yielded a mutually consistent and continuous record of temperature, global ice volume and benthic δ18O over the past 35 Ma. The simple 1-D model shows good agreement with a comprehensive 3-D ice-sheet model for the past 3 Ma. On average, differences are only 1.0˚C for temperature and 6.2 m for sea level. Most notably, over the 35 Ma period, the reconstructed ice volume–temperature sensitivity shows a transition from a climate controlled by Southern Hemisphere ice sheets to one controlled by Northern Hemisphere ice sheets. Although the transient behaviour is important, equilibrium experiments show that the relationship between temperature and sea level is linear and symmetric, providing limited evidence for hysteresis. Furthermore, the results show a good comparison with other simulations of Antarctic ice volume and observed sea level.