Pressure dependence of elastic properties of crystalline Au is calculated based on the first-principles method within the local density approximation for the first time. The calculated equation of state agrees quite well with the results of previous high-pressure experiments. From the present study we find several unique features of the elasticity of Au as follows. (1) The calculated dependence of elastic constants on pressure supports the experiments demonstrating relatively smaller pressure derivatives. (2) The small tetragonal shear constant c(s)=(c(11)-c(12))/2 in Au is remarkably insensitive to pressure, but elastic instability does not appear in the calculated pressure range. This feature is inferred to result from a small covalent nature in the chemical bonding in Au. (3) The elastic anisotropy increases with pressure. The normalized elastic constants considerably deviate from the ideal values and keep almost constant in wide volume variations.