Multi-principal-component alloys have attracted great interest as a novel paradigm in alloy design, with often unique properties and a vast compositional space auspicious for materials discovery. High entropy alloys (HEAs) belong to this class and are being investigated for prospective nuclear applications with reported superior mechanical properties including high-temperature strength and stability compared to conventional alloys. Computational materials design has the potential to play a key role in screening such alloys, yet for high-temperature properties, challenges remain in finding an appropriate balance between accuracy and computational cost. Here we develop an approach based on density-functional theory (DFT) and thermodynamic integration aided by machine learning based interatomic potential models to address this challenge. We systematically evaluate and compare the efficiency of computing the full free energy surface and thermodynamic properties up to the melting point at different stages of the thermodynamic integration scheme. Our new approach provides a ×4 speed-up with respect to comparable free energy approaches at the level of DFT, with errors on high-temperature free energy predictions less than 1 meV/atom. Calculations are performed on an equiatomic HEA, TaVCrW - a low-activation composition and therefore of potential interest for next generation fission and fusion reactors.