  In Newtonian theory, gravity inside a constant density static sphere is
independent of spacetime dimension. Interestingly this general result is also
carried over to Einsteinian as well as higher order Einstein-Gauss-Bonnet
(Lovelock) gravity notwithstanding their nonlinearity. We prove that the
necessary and sufficient condition for universality of Schwarzschild interior
solution describing a uniform density sphere for all $n\geq4$ is that its
density is constant.
