Megadroughts are comparable in severity to the worst droughts of the 20th century but are of much longer duration. A megadrought in the American Southwest would impose unprecedented stress on the limited water resources of the area, making it critical to evaluate future risks not only under different climate change mitigation scenarios but also for different aspects of regional hydroclimate. We find that changes in the mean hydroclimate state, rather than its variability, determine megadrought risk in the American Southwest. Estimates of megadrought probabilities based on precipitation alone tend to underestimate risk. Furthermore, business-as-usual emissions of greenhouse gases will drive regional warming and drying, regardless of large precipitation uncertainties. We find that regional temperature increases alone push megadrought risk above 70, 90, or 99% by the end of the century, even if precipitation increases moderately, does not change, or decreases, respectively. Although each possibility is supported by some climate model simulations, the latter is the most common outcome for the American Southwest in Coupled Model Intercomparison 5 generation models. An aggressive reduction in global greenhouse gas emissions cuts megadrought risks nearly in half.