Although it is widely assumed that the selective advantage of niche specialization drives species biodiversity, some theory suggests that generalists are favored over specialists when environments change unexpectedly. But this idea is rarely tested empirically, and its relevance is unknown for microparasites such as RNA viruses. Due to their small genome sizes pleiotropy is not uncommon in RNA viruses. Therefore, the genetic architectures underlying generalist traits may be indirectly molded by selection to better prepare generalist organisms for growth in new environments. Previously, vesicular stomatitis viruses were evolved to specialize on a single host, or to generalize on multiple hosts. Here we test whether virus generalists arising in the context of host adaptation also perform differently than specialists when viruses grow at novel temperatures. We compared thermal reaction norms of performance, within and among groups of viral specialists and generalists. Results showed that host adaptation was consequential for some fitness traits at novel temperatures due to modification of pleiotropic viral genes. Contrary to theoretical predictions, host generalists were selectively disadvantaged at extreme cool and warm environments. Multi-host adaptation may compromise the evolved thermostability of viral proteins, creating a cost of host generalization when viruses replicate at extreme temperatures.