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Constant environments are often assumed to favor the evolution of specialization whereas exposure to changing environments may favor the evolution of generalists. Here we explored the phenotypic and molecular changes associated with evolving an RNA virus in constant versus fluctuating temperature environments. We used vesicular stomatitis virus (VSV) to determine whether selection at a constant temperature entails a performance trade-off at an unselected temperature, whether virus populations evolve to be generalists when selected in deterministically changing temperature environments, and whether selection under stochastically changing temperatures prevents evolved generalization, such as by constraining the ability for viruses to adaptively improve. We observed that all VSV lineages evolved at constant temperatures showed fitness gains in their selected temperature with little evidence for trade-offs in performance in the unselected environment. Evolution in deterministically and stochastically changing temperatures led to populations with the highest and lowest overall fitness gains, respectively. Sequence analysis revealed little evidence for convergent molecular evolution among lineages within the same treatment. Across all temperature treatments, the majority of genome substitutions occurred in the G (glycoprotein) gene, suggesting that this locus for the cell-binding protein plays a key role in dictating VSV performance under changing temperature.