In this note, we prove three rigidity results for Einstein manifolds with
bounded covering geometry. (1) An almost flat manifold $(M,g)$ must be flat if it is
Einstein, i.e. ${\rm Ric}_g =λg$ for some real number $λ.$ (2) A compact Einstein manifold with
a non-vanishing and almost maximal volume entropy is hyperbolic. (3) A compact
Einstein manifold admitting a uniform local rewinding almost maximal volume is isometric to a space form.
