To study the role of the Atlantic meridional overturning circulation (AMOC) in transient climate change, we perform an abrupt CO2‐doubling experiment using a coupled atmosphere‐ocean‐ice model with a simple geometry that separates the ocean into small and large basins. The small basin exhibits an overturning circulation akin to the AMOC. Over the simulated 200 years of change, it stores heat at a faster rate than the large basin by 0.6 ± 0.2 W m−2. We argue that this is due to the small basin MOC. However, we find that as the MOC weakens significantly, it has little impact on the small basin's heat storage rate. We suggest this is due to the effects of both compensating warming patterns and interbasin heat transports. Thus, although the presence of a MOC is important for enhanced heat storage, MOC weakening is surprisingly unimportant.