  We construct a $\mathcal N$-function for Lovelock theories of gravity, which
yields a holographic $c$-function in domain-wall backgrounds, and seemingly
generalizes the concept for black hole geometries. A flow equation equates the
monotonicity properties of $\mathcal N$ with the gravitational field, which has
opposite signs in the domain-wall and black hole backgrounds, due to the
presence of negative/positive energy in the former/latter, and accordingly
$\mathcal N$ monotonically decreases/increases from the UV to the IR. On $AdS$
spaces the $\mathcal N$-function is related to the Euler anomaly, and at a
black hole horizon it is generically proportional to the entropy. For planar
black holes, $\mathcal N$ diverges at the horizon, which we interpret as an
order $N^2$ increase in the number of effective degrees of freedom. We show how
$\mathcal N$ can be written as the ratio of the Wald entropy to an effective
phase space volume, and using the flow equation relate this to Verlinde's
notion of gravity as an entropic force. From the effective phase space we can
obtain an expression for the dual field theory momentum cut-off, matching a
previous proposal in the literature by Polchinski and Heemskerk. Finally, we
propose that the area in Planck units counts states, not degrees of freedom,
and identify it also as a phase space volume. Written in terms of the proper
radial distance $\beta$, it takes the suggestive form of a canonical partition
function at inverse temperature $\beta$, leading to a "mean energy" which is
simply the extrinsic curvature of the surface. Using this we relate this
definition of holographic phase space with the effective phase space appearing
in the $\mathcal N$-function.
