A substantial difference between the vertical thermal conductance response of homogeneous and heterogeneous graphene/hexagonal boron nitride (h-BN) interfaces to external mechanical perturbations is predicted. Homogeneous graphene and h-BN interfaces exhibit strong conductance reduction for both in-plane strain and interfacial twist. Conversely, the vertical thermal conductance of the heterogeneous graphene/h-BN junction is insensitive to twist deformations but shows significant increase or decrease under compressive or tensile strains, respectively. Our atomistic simulation predictions are rationalized by Fermi's golden rule and density of phonon modes analyses, indicating that vertical phonons and local stacking configurations have a central role in the interlayer heat transport behavior. A simple phenomenological model, based on local interlayer distance and stacking, captures well the dependence of vertical heat conductance on strain and twist deformations.