The apsidal-motion constants k_j_ and the moment of inertia are often used to study the apsidal-motion and tidal evolution of double-lined eclipsing binaries and planetary systems. On the other hand, the computation of the theoretical light curves of eclipsing binaries, planetary transits, and single rapidly rotating stars requires the knowledge of how the flux is distributed over the distorted stellar surfaces which can derived from the gravity-darkening exponent. Such parameters are available for several masses and chemical compositions covering the main-sequence and giant branch. However, for early phases (pre main-sequence, hereafter PMS) the calculations are scarce or even lacking. We present the calculations of the apsidal-motion constants, the fractional radius of gyration, and the gravity-darkening exponents for an extensive grid of PMS. The code used to generate the PMS models is essentially the same as that described by us in 2004. The apsidal-motion constants, the moment of inertia and the potential energy were computed using a fourth order Runge-Kutta method. The gravity-darkening exponents were computed using a method previously developed by us. The apsidal-motion constants, the moment of inertia, the potential energy, and the gravity-darkening exponents are made available for each point on every evolutionary track for PMS models covering the mass range 0.05-30M_{sun}_. Our calculations are made available for three chemical compositions (X, Z)=(0.757, 0.001), (0.70, 0.02), (0.64, 0.04).