Grain boundaries (GBs), the most important defects in solids and their properties are crucial for many materials properties including (in-)stability. Quantum-mechanical methods can reliably compute properties of GBs and we use them to analyze (tensorial) anisotropic elastic properties of interface states associated with GBs in one of the most important intermetallic compounds for industrial applications, Ni3Al. Selecting the Σ5(210) GBs as a case study because of its significant extra volume, we address the mechanical stability of the GB interface states by checking elasticity-based Born stability criteria. One critically important elastic constant, C55, is found nearly three times smaller at the GB compared with the bulk, contributing thus to the reduction of the mechanical stability of Ni3Al polycrystals. Next, comparing properties of Σ5(210) GB state which is fully relaxed with those of a Σ5(210) GB state when the supercell dimensions are kept equal to those in the bulk we conclude that lateral relaxations have only marginal impact on the studied properties. Having the complete elastic tensor of Σ5(210) GB states we combine Green's-function based homogenization techniques and an approximative approach to the Debye model to compare thermodynamic properties of a perfect Ni3Al bulk and the Σ5(210) GB states. In particular, significant reduction of the melting temperature (to 79-81% of the bulk value) is predicted for nanometer-size grains.