  We study some universal features of gravity in higher dimensions and by
universal we mean a feature that remains true in all dimensions $\geq4$. They
include: (a) the gravitational dynamics always follows from the Bianchi
derivative of a homogeneous polynomial in Riemann curvature and it thereby
characterizes the Lovelock polynomial action, (b) all the $\Lambda$-vacuum
solutions of the Einstein-Lovelock as well as pure Lovelock equation have the
same asymptotic limit agreeing with the $d$ dimensional Einstein solution and
(c) gravity inside a uniform density sphere is independent of the spacetime
dimension and it is always given by the Schwarzschild interior solution.
