The effects of climate change could vary across the income distribution. Focusing on the case of extreme temperatures in the US, this paper develops an Aiyagari-style heterogenous agent model to study the distributional consequences of climate change when households face idiosyncratic income shocks. Households can adapt to extreme temperatures by using energy for heating and cooling. The model replicates the empirical relationships between energy budget shares, energy expenditures, and income. I find that extreme temperatures have higher welfare costs for lower income households. Consequently, climate change is regressive in hot regions of the US, where it leads to more extreme temperatures, and progressive in cold regions, where it leads to fewer extreme temperatures. US energy assistance policy reduces but does not eliminate these distributional effects. Accounting for the distributional impacts of climate change substantially alters the aggregate welfare consequences within a region.