It remains unclear how COVID-19 has affected public engagement with the climate crisis. According to the finite-pool-of-worry hypothesis, concern about climate change should have decreased after the pandemic, in turn reducing climate-policy acceptance. Here we test these and several other conjectures by using survey data from 1172 Spanish participants who responded before and after the first wave of COVID-19, allowing for both aggregate and within-person analyses. We find that on average climate concern has decreased, while acceptance of most climate policies has increased. At the individual-level, adverse health experiences are unrelated to these changes. The same holds for negative economic experiences, with the exception that unemployment is associated with reduced acceptance of some policies. Complementary to the finite-pool-of-worry test, we examine three additional pandemic-related issues. As we find, (1) higher climate concern and policy acceptance are associated with a belief that climate change contributed to the COVID-19 outbreak; (2) higher policy acceptance is associated with a positive opinion about how the government addressed the COVID-19 crisis; (3) citizens show favorable attitudes to a carbon tax with revenues used to compensate COVID-19-related expenditures. Overall, we conclude there is support for addressing the global climate crisis even during a global health crisis.