•Elastic properties of titanium borides are calculated by first principles calculation.•Thermodynamical stability of titanium borides is analyzed.•Heat capacity and thermal expansion coefficient for titanium borides are calculated and compared.•Grüneisen parameters of titanium borides are calculated. The anisotropic elastic and thermal expansions of the titanium borides (TiB2, Ti3B4, TiB_Pnma and TiB_Fm3¯m) are calculated from first-principles using density functional theory. All borides show different anisotropic elastic properties; the bulk, shear and Young’s moduli are consistent with those determined experimentally. The temperature dependence of thermal expansions is mainly caused by the restoration of thermal energy due to phonon excitations at low temperature. When the temperature is higher than 500K, the volumetric coefficient is increased linearly by increasing temperature. Meanwhile, the heat capacities of titanium borides are obtained based on the knowledge of thermal expansion coefficient and the elasticity, the calculations are in good agreement with the experiments.