The response of extreme precipitation to warming varies widely among climate models, especially in the tropics. In some models, there have been indications that the rate of response increases with warming—that the response is not linear. We investigate the evolution of extreme precipitation, quantified by the maximum accumulated precipitation in a day each year, in CESM1. We find that tropical‐ and global‐average extreme precipitation is related to global‐mean surface temperature quadratically. This behavior is associated with an increase in the large‐scale fraction of extreme precipitation and also strengthening circulation on extreme precipitation days. Compared to other CMIP5 models, the nonlinearity in CESM1 is among the largest. One implication is that the difference between CESM1 simulations with full forcing and with fixed aerosols cannot be used to isolate the response of extreme precipitation to aerosols, as the resulting climates are not equally warm. Plain Language Summary: Extreme precipitation can drive natural disasters like floods and landslides, so understanding and quantifying how it responds to warming is important. Climate models disagree on how much extreme precipitation changes in response to global warming in the tropics. Here we focus on trying to understand the response in just one climate model, CESM1. Some previous studies using this model assumed that the response of extreme precipitation for a given amount of warming is fixed. This is what we would expect if the change in extreme precipitation followed the amount of moisture in the atmosphere. However, we find that the response of extreme precipitation in the tropics to a given amount of warming is not fixed, and instead increases as the temperature warms. Changes in circulation and in the way the model produces precipitation accompany this behavior. Among other climate models, this behavior is shared by some but not all models. CESM1 has a larger increase in extreme precipitation change in response to warming than most models, and some lack this behavior entirely. The next generation of models that descend from CESM1 also do not share this behavior. Key Points: In CESM1, extreme precipitation (the heaviest day each year) is quadratically related to warming in the tropicsExtreme precipitation change is closely related to circulation strength and large‐scale precipitation fractionCESM1 is an end member among its cohort of climate models for this behavior [ABSTRACT FROM AUTHOR]