Unfortunately I'm not very familiar with the proof methods of the laws of black hole mechanics/thermodynamics. Do they hold in any number of dimensions? Are there any "hidden" assumptions? I'm thinking natural assumptions like asymptotic flatness (which would not hold for AdS black holes), or cosmic censorship (which I believe is violated in $D\geq 5$). Or are the laws simply general statements about event horizons?
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The mass-energy, radius, entropy, lifetime etc. are related with power laws whose exponents depend on not only $D$ but in some cases also the geometry. For example, which $p$ gives the entropy-energy relation $S\propto E^p$? If the universe is otherwise Minkowski, $p=\frac{D}{D-2}$; if it's anti-de Sitter, $p=\frac{D-2}{D-1}$. As explained here, the former indicates quantum gravity is non-renormalisable while the latter motivates interest in the AdS/CFT correspondence.
