  We study the imaginary part of the Lorentzian gravitational action for
bounded regions, as described in arXiv:1301.7041. By comparing to a Euclidean
calculation, we explain the agreement between the formula for this imaginary
part and the formula for black hole entropy. We also clarify the topological
structure of the imaginary part in Lovelock gravity. We then evaluate the
action's imaginary part for some special regions. These include cylindrical
slabs spanning the exterior of a stationary black hole spacetime, 'maximal
diamonds' in various symmetric spacetimes, as well as local near-horizon
regions. In the first setup, the black hole's entropy and conserved charges
contribute to the action's imaginary and real parts, respectively. In the other
two setups, the imaginary part coincides with the relevant entropy.
