The exact expression for the exchange-correlation energy functional Exc[ρ,jp] of the current-density functional theory (CDFT) is derived by means of the coupling-constant integration technique. It contains the coupling-constant-averaged pair correlation function, which is a functional of the electron density ρ(r) and paramagnetic current density jp(r). On the basis of this expression, the local density approximation and its modifications for Exc[ρ,jp] are proposed within the CDFT. In addition, we present sum rules and bounds on Exc[ρ,jp] by considering the behaviors of the basic variables and Exc[ρ,jp] under the various types of the nonuniform coordinate scaling of electrons. They are useful in estimating the validity of the approximate forms of Exc[ρ,jp] proposed.