  Entanglement entropies calculated in the framework of quantum field theory on
classical, flat or curved, spacetimes are known to show an intriguing area law
in four dimensions, but they are also notorious for their quadratic ultraviolet
divergences. In this paper we demonstrate that the analogous entanglement
entropies when computed within the Asymptotic Safety approach to background
independent quantum gravity are perfectly free from such divergences. We argue
that the divergences are an artifact due to the over-idealization of a rigid,
classical spacetime geometry which is insensitive to the quantum dynamics.
