The elastic and thermodynamic properties of the anti-perovskite superconductor ZnCNi_3 and CdCNi_3 are investigated by first-principles calculations. With the local density approximation as well as the generalized gradient approximation for exchange and correlation, the ground state properties and equation of state are obtained, which agree well with other theoretical calculations and experiments. Furthermore, by the elastic stability criteria, we predict that ZnCNi_3 and CdCNi_3 are not stable above 98.1 GPa and 196.5 GPa, respectively. The dependences of the heat capacity, thermal expansion coefficient, the Grüneisen parameter and bulk modulus (B_{T} and B_{S}) on pressure and temperature for ZnCNi_3 and CdCNi_3 are also obtained successfully.
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