Since the origin of life, temperatures on earth have fluctuated both on short and long time scales. How such changes affect the rate at which Darwinian evolution can bring forth new phenotypes remains unclear. On the one hand, high temperature may accelerate phenotypic evolution because it accelerates most biological processes. On the other hand, it may slow phenotypic evolution, because proteins are usually less stable at high temperatures and therefore less evolvable. Here, to test these hypotheses experimentally, we evolved a green fluorescent protein in E. coli towards the new phenotype of yellow fluorescence at different temperatures. Yellow fluorescence evolved most slowly at high temperature and most rapidly at low temperature, in contradiction to the first hypothesis. Using high-throughput population sequencing, protein engineering, and biochemical assays, we determined that this is due to the protein-destabilizing effect of neofunctionalizing mutations. Destabilization is highly detrimental at high temperature, where neofunctionalizing mutations cannot be tolerated. Their detrimental effects can be mitigated through excess stability at low temperature, leading to accelerated adaptive evolution. By modifying protein folding stability, temperature alters the accessibility of mutational paths towards high-fitness genotypes. Our observations have broad implications for our understanding of how temperature changes affect evolutionary adaptations and innovations.