Investigating the history of natural selection among closely related species can elucidate how genomes diverge in response to disparate environmental pressures. Molecular evolutionary approaches can be integrated with knowledge of gene functions to examine how evolutionary divergence may affect ecologically-relevant traits such as temperature tolerance and species distribution limits. Here, we integrate transcriptome-wide analyses of molecular evolution with knowledge from physiological studies to develop hypotheses regarding the functional classes of genes under positive selection in one of the world’s most widespread invasive species, the warm-tolerant marine mussel Mytilus galloprovincialis. Based on existing physiological information, we test the hypothesis that genomic functions previously linked to divergent temperature adaptation at the whole-organism level show accelerated molecular divergence between warm-adapted M. galloprovincialis and cold-adapted congeners. Combined results from codon model tests and analyses of polymorphism and divergence reveal that divergent selection has affected genomic functions previously associated with species-specific expression responses to heat stress, namely oxidative stress defense and cytoskeletal stabilisation. Examining specific loci implicated in thermal tolerance among Mytilus species (based on interspecific biochemical or expression patterns), we find close functional similarities between known thermotolerance candidate genes under positive selection and positively selected loci under predicted genomic functions (those associated with divergent expression responses). Taken together, our findings suggest a contribution of temperature-dependent selection in the molecular divergence between warm- and cold-adapted Mytilus species that is largely consistent with results from physiological studies. More broadly, this study provides an example of how independent experimental evidence from ecophysiological investigations can inform evolutionary hypotheses about molecular adaptation in closely related non-model species.