Higher-Dimensional Black Hole Hawking Temperature and Entropy from the RVB--Residue Method:Tangherlini--AdS and Einstein--Gauss--Bonnet Examples

We formulate a dimensionally explicit local Robson--Villari--Biancalana (\RVB) residue prescription for the Hawking temperature and entropy of higher-dimensional black holes. For a nonextremal horizon of the $D$-dimensional static metric $\dd s^2=-A(r)\dd t^2+\dd r^2/B(r)+r^2\dd\Omega_{D-2}^2$, with $A$ and $B$ analytic near the horizon, the locally meromorphic one-form \[ \omega_h=\frac{\dd r}{\sqrt{A(r)B(r)}} \] has a simple pole. Its positive residue $\mathscr{R}_h=|\Res_{r=r_h}\omega_h|$ gives $\beta_h=4\pi\mathscr{R}_h$ and $T_h=(4\pi\mathscr{R}_h)^{-1}$. The Laurent coefficient and Euclidean regularity establish this relation; the contour representation and the two-dimensional Gauss--Bonnet formula connect it with the original \RVB\ construction. The relevant Euler characteristic is that of the Euclidean $(\tau,r)$ orbit-space disk, not that of the $(D-2)$-dimensional horizon cross-section. When work variables and couplings are held fixed, combining the inverse temperature with the first law gives \[ \frac{\dd S}{\dd\lambda} =4\pi\mathscr{R}_h(\lambda)\frac{\dd M}{\dd\lambda}. \] For the $D$-dimensional Schwarzschild--Tangherlini--AdS family this procedure gives \[ T_h=\frac{1}{4\pi}\left(\frac{D-3}{r_h} +\frac{(D-1)r_h}{\ell^2}\right), \qquad S_h=\frac{\Omega_{D-2}r_h^{D-2}}{4G_D}. \] For the Einstein branch of spherical Einstein--Gauss--Bonnet--AdS black holes in $D\geq5$, it gives \[ T_h= \frac{(D-3)r_h^2+(D-5)\widetilde{\alpha} +(D-1)r_h^4/\ell^2} {4\pi r_h(r_h^2+2\widetilde{\alpha})}, \] \[ S_h=\frac{\Omega_{D-2}r_h^{D-2}}{4G_D} \left[ 1+\frac{2(D-2)\widetilde{\alpha}}{(D-4)r_h^2} \right]. \] The latter expression agrees with the Wald--Jacobson--Myers entropy. We also check the Einstein $D=4$ limit, the zero-coupling and asymptotically flat limits, dimensional consistency, and a numerical finite-difference reproduction. The construction is local and fixed-coupling: degenerate horizons, varying pressure or Gauss--Bonnet coupling, and globally nonequilibrium multi-horizon temperatures require separate treatments.