Significance Assessing the response of clouds to global warming remains a challenge of climate science. Past research has elucidated what controls the height and temperature of high-level anvil clouds, but the factors that control their horizontal extent remained uncertain. We show that the anvil cloud amount is expected to shrink as the climate warms or when convection becomes more clustered, due to a mechanism rooted in basic energetic and thermodynamic properties of the atmosphere. It is supported by three climate models and consistent with results from a cloud-resolving model and observations. We thus believe that this mechanism is robust and that it adds a new piece to understanding how clouds respond to climate warming.