Cross-societal differences in cooperation and trust among strangers in the provision of public goods may be key to understanding how societies are managing the COVID-19 pandemic. We report a survey conducted across 41 societies between March and May 2020 (N = 34,526), and test pre-registered hypotheses about how cross-societal differences in cooperation and trust relate to prosocial COVID-19 responses (e.g., social distancing), stringency of policies, and support for behavioral regulations (e.g., mandatory quarantine). We further tested whether cross-societal variation in institutions and ecologies theorized to impact cooperation were associated with prosocial COVID-19 responses, including institutional quality, religiosity, and historical prevalence of pathogens. We found substantial variation across societies in prosocial COVID-19 responses, stringency of policies, and support for behavioral regulation. However, we found no consistent evidence to support the idea that cross-societal variation in cooperation and trust among strangers is associated with these outcomes related to the COVID-19 pandemic. These results were replicated with another independent cross-cultural COVID-19 dataset (N= 112,136), and in both snowball and representative samples. We discuss implications of our results, including challenging the assumption that managing the COVID-19 pandemic across societies is best modeled as a public goods dilemma.