We calculate communication maps for green fluorescent protein (GFP) to elucidate energy transfer pathways between the chromophore and other parts of the protein in the ground and excited state. The approach locates energy transport channels from the chromophore to remote regions of the protein via residues and water molecules that hydrogen bond to the chromophore. We calculate the thermal boundary conductance between GFP and water over a wide range of temperature and find that the interface between the protein and the cluster of water molecules in the beta-barrel poses negligible resistance to thermal flow, consistent with facile vibrational energy transfer from the chromophore to the beta-barrel waters observed in the communication maps.