The atomistic mechanisms of deformation in multicomponent random alloys are challenging to model because of their extensive structural and compositional disorder. For embedded-atom-method interatomic potentials, a formal averaging procedure can generate an average-atom EAM potential and this average-atom potential has recently been shown to accurately predict many zero-temperature properties of the true random alloy. Here, the finite-temperature thermodynamic properties of the average-atom potential are investigated to determine if the average-atom potential can represent the true random alloy Helmholtz free energy as well as important finite-temperature properties. Using a thermodynamic integration approach, the average-atom system is found to have an entropy difference of at most 0.05 k B/atom relative to the true random alloy over a wide temperature range, as demonstrated on FeNiCr and Ni85Al15 model alloys. Lattice constants, and thus thermal expansion, and elastic constants are also well-predicted (within a few percent) by the average-atom potential over a wide temperature range. The largest differences between the average atom and true random alloy are found in the zero temperature properties, which reflect the role of local structural disorder in the true random alloy. Thus, the average-atom potential is a valuable strategy for modeling alloys at finite temperatures.