In this work, mechanical stability, thermodynamic and elastic properties of rhodium (Rh), rhodium monohydride (RhH), and the newly discovered rhodium dihydride (RhH2) under high temperature and pressure are studied by ab initio method together with quasiharmonic Debye model. Mechanical stability test indicates that RhH2 is no longer mechanically stable when pressure is higher than 22.7 GPa, which is quite less than the dynamically stable pressure (90 GPa). The heat capacity at constant volume (Cv) of Rh, RhH, or RhH2 increases proportional to T3 at low temperature, and tends to Dulong–Petit limit (about 241.67, 478.47, and 706.15 J/(kg·K), respectively). The thermal expansion coefficient (α) of Rh, RhH, and RhH2 increases acutely when temperature is not more than 300 K. And then, the increase of α slows down. The α reduces with pressure transiently. H atom's entering in fcc-Rh lattice would greatly change the electron density distribution, which would cause obvious difference in thermodynamic and elastic properties between Rh, RhH, and RhH2.