We demonstrate a theoretically expected behavior of the tropical sea surface temperature probability density function (PDF) in future and past (Eocene) greenhouse climate simulations. To first order this consists of a shift to warmer temperatures as climate warms, without change of shape of the PDF. The behavior is tied to a shift of the temperature for deep convection onset. Consequently, the threshold for appearance of high clouds and associated radiative forcing shifts along with temperature. An excess entropy coordinate provides a reference to which the onset of deep convection is invariant, and gives a compact description of SST changes and cloud feedbacks suitable for diagnostics and as a basis for simplified climate models. The results underscore that the typically skewed appearance of tropical SST histograms, with a sharp drop-off above some threshold value, should not be taken as evidence for tropical thermostats.