  As observed recently in arXiv:1212.2922, the action of General Relativity
(GR) in finite spacetime regions has an imaginary part that resembles the
Bekenstein entropy. In this paper, we expand on that argument, with attention
to different causal types of boundaries. This property of the GR action may
open a new approach to the puzzles of gravitational entropy. In particular, the
imaginary action can be evaluated for non-stationary finite regions, where the
notion of entropy is not fully understood. As a first step in constructing the
precise relationship between the imaginary action and entropy, we focus on
stationary black hole spacetimes. There, we identify a class of bounded regions
for which the action's imaginary part precisely equals the black hole entropy.
As a powerful test on the validity of the approach, we also calculate the
imaginary action for Lovelock gravity. The result is related to the
corresponding entropy formula in the same way as in GR.
