Many practical flow configurations involve energy transfer in fluids, or in solids and fluids with different thermo-physical properties. The classical advection-diffusion lattice Boltzmann (LB) solver admits some errors when dealing with such configurations. Given that the macroscopic equation recovered by this model is only valid in the limit of incompressible flows with constant heat capacities, one would, for example, observe inconsistent fluxes at the interface of a fluid and solid with different densities or specific heat capacities. This inconsistency being second-order in space, it will have non-negligible effects on the final results. In this work, a modified equilibrium distribution function (EDF) is proposed to overcome these issues. The proposed scheme recovers the correct partial differential equation (PDE) describing energy transfer, as shown by a multi-scale Chapman-Enskog analysis. The performance of the model is checked through a variety of test-cases, involving conjugate heat transfer and variable specific heat capacities in both steady and unsteady configurations. In all cases the obtained results are in excellent agreement with reference data.