Global warming simulations robustly show that mean precipitation increases at 1-3% per Kelvin, but we do not know what sets these values. Mean precipitation is constrained by radiative cooling, however, and we demonstrate here that radiative cooling profiles exhibit a certain invariance under warming when plotted in temperature coordinates. This invariance can then be leveraged to derive simple analytical equations for precipitation change with warming. These equations are validated in cloud-resolving simulations of the tropics, and give intuition for why precipitation changes at a rate of 1-3% per Kelvin. We also discuss the application of this formalism to more comprehensive GCM output.