Computational methods that characterize the thermodynamic properties of metal hydrides that operate at high temperatures, i.e., T > 800 K, are desirable for a variety of applications, including nuclear fuels and energy storage. Ternary hydrides tend to be less thermodynamically stable than the strongest binary hydride that forms from the metals. In this paper we use first-principles methods based on density functional theory, phonon calculations, and grand potential minimization to predict the isobaric phase diagram for 0 K <= T <= 2000 K for the ThZrH element space, which is of interest given that ThZr2Hx ternary hydrides have been reported with an enhanced stability relative to the binary hydrides. We compute free energies including vibrational contributions for Th, Zr, ThH2, Th4H15, ZrH2, ThZr2, ThZr2H6, and ThZr2H7. We develop a cluster analysis method that efficiently computes the configurational entropy for ThZr2H6 interstitial hydride and conclude that the configurational entropy is not a major driver for the enhanced stability of ThZr2H6 relative to the binary hydrides. Density functional theory (DFT) predicted thermodynamic stabilities for the hydride phases are in reasonable agreement with experimental values. ThZr2H6 is stabilized by finite temperature vibrational effects, and ThZr2H7 is not predicted to be stable at any studied temperature or pressure.