Understanding the reasons for which current climate models fail to reproduce the low equator-to-pole temperature gradient of past warm periods is among the major challenges in climate science. We focus here on the role of atmospheric poleward heat transport. We use an aquaplanet general circulation model (GCM) to construct a regime diagram of this quantity as a function of surface temperature and its meridional gradient, encompassing the range experienced by Earth over the Cenozoic. We find that poleward heat transport increases with surface temperature over much of this range, but saturates in the low-gradient, high-temperature regime where it is most needed. We identify some specific dynamical feedbacks responsible for this behavior: increasing tropospheric static stability and poleward migration of the storm tracks as global-mean temperature increases.