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The invariant four-volume $\mathcal{V}$ of a complete black hole (the volumeof the spacetime at and interior to the horizon) diverges. However, if oneconsiders the black hole set up by the gravitational collapse of an object, andintegrates only a finite time to the future of the collapse, the resultantvolume is well defined and finite. In this paper we examine non-degeneratestationary black holes (and cosmological horizons) and find that$\mathcal{V}_{s} \varpropto \ln(\lambda)$ where $s$ is any shell thatterminates on the horizon, $\lambda$ is the affine generator of the horizon andthe constant of proportionality is the Parikh volume of $s$ divided by thesurface gravity. This provides an alternative local and invariant definition ofthe surface gravity of a stationary black hole.

The volume of stationary black holes and the meaning of the surface gravity