The elastic anisotropy and thermodynamic properties of the potential superhard carbon nitride phase C2N2(NH) have been investigated by using an ab initio plane-wave pseudopotential density theory method. The crystal parameters have been calculated at ambient as well as high pressure. The Young's modulus and shear modulus as a function of crystal orientations for C2N2(NH) have been systematically investigated. The Young's modulus is found to reach a maximum along the  direction. Using a set of total energy versus volume obtained with the first-principles calculations, the quasiharmonic Debye model is applied to the study of the thermal and vibrational effects. The dependence of Debye temperature, Grüneisen parameter, heat capacity, and expansion coefficient on the temperature and pressure are systematically explored in the whole pressure range from 0 to 60 GPa and temperature range from 0 to 2000 K.