Mounting evidence suggests that rapid evolutionary adaptation may rescue some organisms from the impacts of ongoing climate change. However, evolutionary constraints might hinder this process, especially when different aspects of environmental change generate antagonistic selection on genetically correlated traits. Here, we use individual-based simulations to explore how genetic correlations underlying the thermal physiology of ectotherms might influence their responses to the two major concomitant components of climate change---increases in mean temperature and thermal variability. We found that genetic correlations can influence population dynamics under climate change, with declines in population size varying three-fold depending on the type of correlation present. Surprisingly, populations whose thermal performance curves were constrained by genetic correlations often declined less rapidly than unconstrained populations. Our results suggest that accurate forecasts of the impact of climate change on ectotherms will require an understanding of the genetic architecture of the traits under selection.