We study heat transport in hybrid lateral normal-metal-superconductor-normal-metal structures. We find the thermal conductance of a short superconducting wire to be strongly enhanced beyond the BCS value due to the inverse proximity effect, resulting from contributions of elastic cotunneling and crossed Andreev reflection of quasiparticles. Our measurements agree with a model based on the quasiclassical theory of inhomogeneous superconductivity in the diffusive limit.