We present the graviton propagator for an infinite derivative, D-dimensional, nonlocal action, up to quadratic order in curvature around a Minkowski background, and discuss the conditions required for this class of gravity theory to be ghost-free. We then study the gravitational entropy for de-Sitter and anti-de Sitter backgrounds, before comparing with a recently derived result for a Schwarzschild black hole, generalized to arbitrary D dimensions, whereby the entropy is given simply by the area law. A novel approach of decomposing the entropy into its (r,t) and spherical components is adopted in order to illustrate the differences more clearly. We conclude with a discussion of de-Sitter entropy in the framework of a nonsingular bouncing cosmology.