The Finite Pool of Worry (FPW) hypothesis states that humans have finite emotional resources for worry, so that when we become more worried about one threat, it can decrease worry about other threats. Despite its relevance, no conclusive empirical evidence for the hypothesis exists. We leverage the sudden onset of new worries introduced by the COVID-19 pandemic as a natural experiment to test the FPW. In six metropolitan areas across three countries (USA, Italy, and China) we assessed social media attention, news attention, self-reported attention, and self-reported worries about various threats (climate change, terrorism, economy, and unemployment) throughout the pandemic. As attention to and worry about COVID-19 increased, we find that attention to climate change decreased but worry about it did not. Results are confirmed by further analysis with a large, and nationally representative U.S. sample. We find some perceived similarity between COVID-19 and climate change, but this does not fully explain the positive relationship in worry we see between them. We also find that more negative personal experience with COVID-19 is positively associated with climate change worry even while controlling for relevant covariates. We lastly examine the aggregate effect of COVID-19 worry on support for climate policies and find that greater COVID-19 worry is associated with more cross-partisan support for climate change policies, even when controlling for political ideology and other covariates. In summary, our findings suggest that while there appears to be a Finite Pool of Attention to threats, we do not see evidence of a Finite Pool of Worry.