Energy processes and vibrations in biological macromolecules such as proteins ultimately dictate biological, chemical, and physical functions in living materials. These energetic vibrations in the ribbon-like motifs of proteins interact on self-similar structures and fractal-like objects over a range of length scales of the protein (a few angstroms to the size of the protein itself, a few nanometers). In fact, the fractal geometries of protein molecules create a complex network of vibrations; therefore, proteins represent an ideal material system to study the underlying mechanisms driving vibrational thermal transport in a dense, fractal network. However, experimental studies of thermal energy transport in proteins have been limited to dispersive protein suspensions, which limits the knowledge that can be extracted about how vibrational energy is transferred in a pure protein solid. We overcome this by synthesizing solid, water-insoluble protein films for thermal conductivity measurements via time-domain thermoreflectance. We measure the thermal conductivity of bovine serum albumin and myoglobin solid films over a range of temperatures from 77 to 296 K. These temperature trends indicate that anharmonic coupling of vibrations in the protein is contributing to thermal conductivity. This first-ever observation of anharmonic-like trends in the thermal conductivity of a fully dense protein forms the basis of validation of seminal theories of vibrational energy-transfer processes in fractal objects.