We examine the tropical inversion strength, measured by the Estimated Inversion Strength (EIS), and its response to climate change in 18 models associated with phase 5 of the Coupled Model Intercomparison Project (CMIP5). While CMIP5 models generally capture the geographic distribution of observed EIS, they systematically underestimate it off the west coasts of continents, due to a warm bias in sea surface temperature. The negative EIS bias may contribute to the low bias in tropical low-cloud cover in the same models. Idealized perturbation experiments reveal that anthropogenic forcing leads directly to EIS increases, independent of "temperature-mediated" EIS increases associated with long-term oceanic warming. This fast EIS response to anthropogenic forcing is strongly impacted by nearly instantaneous continental warming. The temperature-mediated EIS change has contributions from both uniform and non-uniform oceanic warming. The substantial EIS increases in uniform oceanic warming simulations are due to warming with height exceeding the moist adiabatic lapse rate in tropical warm pools. EIS also increases in fully-coupled ocean-atmosphere simulations where CO2 concentration is instantaneously quadrupled, due to both fast and temperature-mediated changes. The temperature-mediated EIS change varies with tropical warming in a nonlinear fashion: The EIS change per degree tropical warming is much larger in the early stage of the simulations than in the late stage, due to delayed warming in the eastern parts of the subtropical oceans. Given the importance of EIS in regulating tropical low-cloud cover, this suggests that the tropical low-cloud feedback may also be nonlinear.