In a combined experimental-theoretical study, we investigated the transport of vibrational energy from the surrounding solvent into the interior of a heme protein, the sperm whale myoglobin double mutant L29W-S108L, and its dependence on temperature from 20 to 70 K. The hindered libration of a CO molecule that is not covalently bound to any part of the protein but is trapped in one of its binding pockets (the Xe4 pocket) was used as the local thermometer. Energy was deposited into the solvent by IR excitation. Experimentally, the energy transfer rate increased from (30 ps)(-1) at 20 K to (8 ps)(-1) at 70 K. This temperature trend is opposite to what is expected, assuming that the mechanism of heat transport is similar to that in glasses. In order to elucidate the mechanism and its temperature dependence, nonequilibrium molecular dynamics (MD) simulations were performed, which, however, predicted an essentially temperature-independent rate of vibrational energy flow. We tentatively conclude that the MD potentials overestimate the coupling between the protein and the CO molecule, which appears to be the rate-limiting step in the real system at low temperatures. Assuming that this coupling is anharmonic in nature, the observed temperature trend can readily be explained.