Two‐year simulation experiments with a superparameterized climate model, SP‐CAM, are performed to understand the fast tropical (30S‐30N) cloud response to an instantaneous quadrupling of CO2 concentration with SST held fixed at present‐day values. The greenhouse effect of the CO2 perturbation quickly warms the tropical land surfaces by an average of 0.5 K. This shifts rising motion, surface precipitation, and cloud cover at all levels from the ocean to the land, with only small net tropical‐mean cloud changes. There is a widespread average reduction of about 80 m in the depth of the trade inversion capping the marine boundary layer (MBL) over the cooler subtropical oceans. One apparent contributing factor is CO2‐enhanced downwelling longwave radiation, which reduces boundary‐layer radiative cooling, a primary driver of turbulent entrainment through the trade inversion. A second contributor is a slight CO2‐induced heating of the free troposphere above the MBL, which strengthens the trade inversion and also inhibits entrainment. There is a corresponding downward displacement of MBL clouds with a very slight decrease in mean cloud cover and albedo. Two‐dimensional cloud‐resolving model (CRM) simulations of this MBL response are run to steady state using composite SP‐CAM simulated thermodynamic and wind profiles from a representative cool subtropical ocean regime, for the control and 4xCO2 cases. Simulations with a CRM grid resolution equal to that of SP‐CAM are compared with much finer resolution simulations. The coarse‐resolution simulations maintain a cloud fraction and albedo comparable to SP‐CAM, but the fine‐resolution simulations have a much smaller cloud fraction. Nevertheless, both CRM configurations simulate a reduction in inversion height comparable to SP‐CAM. The changes in low cloud cover and albedo in the CRM simulations are small, but both simulations predict a slight reduction in low cloud albedo as in SP‐CAM.