The structural and elastic properties of the oxide perovskite HoMnO3 have been investigated by density-functional theory (DFT) implemented on the Quantum ESPRESSO code for the hexagonal, orthorhombic, rhombohedral and cubic crystal structures in the non-magnetic (NM), ferromagnetic (FM) and antiferromagnetic (AFM) configurations. The results show that all these compounds are thermodynamically stable, the hexagonal and the orthorhombic being the most stables in agreement with experimental results. The calculated bulk modulus, B, is about 168.5 ± 1.0 GPa and 178.9 ± 1.0 GPa for hexagonal structure in FM and configurations, respectively, and 188.7 ± 0.6 GPa and 175.6 ± 2.1GPa for orthorhombic structure in FM and AFM configurations, respectively. These values and those of the elastic constants Cij in the present work are comparable to values recently obtained with DFT calculations and experimentally for other perovskites RMnO3 (R = Eu, Gd, Tb and Dy). The calculated Poisson’s ratio is about 0.19; 0.27; 0.23 and 0.3 for hexagonal, orthorhombic, rhombohedral and cubic structure, respectively. The calculated Pugh’s ratio is about 1.30; 1.77–1.98; 1.54–1.60 and 2.11, respectively. These results show that the perovskite HoMnO3 tends to be brittle in hexagonal phase and ductile in cubic phase. In the orthorhombic or rhombohedral phase, it is near brittle-ductile boarder line.