Determining the forces that conserve amino acid positions in proteins across species is a fundamental pursuit of molecular evolution. Evolutionary conservation is driven by either a protein's function or its thermodynamic stability. Highly conserved histone proteins offer a platform to evaluate these driving forces. While the conservation of histone H3 and H4 "tail" domains and surface residues are driven by functional importance, the driving force behind the conservation of buried histone residues has not been examined. Using a computational approach, we determined the thermodynamically preferred amino acids at each buried position in H3 and H4. In agreement with what is normally observed in proteins, we find a significant correlation between thermodynamic stability and evolutionary conservation in the buried residues in H4. In striking contrast, we find that thermodynamic stability of buried H3 residues does not correlate with evolutionary conservation. Given that these H3 residues are not post-translationally modified and only regulate H3-H3 and H3-H4 stabilizing interactions, our data imply an unknown function responsible for driving conservation of these buried H3 residues.