  A treatment of linearized gravity with the Einstein-Gauss-Bonnet interaction
terms in $D\geq$ 5 is given in the Randall-Sundrum brane background. This
Letter has outlined some interesting features of the brane world gravity and
Newtonian potential correction with the GB interaction term. We find that the
GB coupling $\alpha$ renormalizes the effective four-dimensional Newton
constant on the brane, and also additionally contributes to the correction term
of the Newtonian potential. Indeed, the GB term does not affect the massless
graviton mode and the Einstein gravity on the brane, and quite interestingly,
such term in $D\geq 5$ appears to give more information about the necessary
boundary condition(s) to be satisfied by the zero mode wavefunction on the
brane(s). small (and preferably positive) GB coupling $\alpha$.
