Equilibrium climate sensitivity (ECS) is a widely accepted measure of Earth's susceptibility to radiative forcing. While ECS is often assumed to be constant to a first order of approximation, recent studies suggested that ECS might depend on the climate state. Here it is shown that the latest generation of climate models consistently exhibits an increasing ECS in warmer climates due to a strengthening of the water-vapor feedback with increasing surface temperatures. The increasing ECS is replicated by a one-dimensional radiative-convective equilibrium model, which further shows that the enhanced water-vapor feedback follows from the rising of the tropopause in a warming climate. This mechanism is potentially important for understanding both warm climates of Earth's past and projections of future high-emission scenarios.