metric

\[ ds^2=-f(r) dt^2 +\frac{1}{f(r)}dr^2+ Ad\theta^2+ Bd\phi^2 +OtherParts\]

event horizon

\[from \  \  f(r)=0,\  \   get \  \  r=r_h\]

surface gravity

\[\kappa=\frac{1}{2 r_h}\]

temperture

\[T=\frac{\kappa}{2 \pi}=\frac{1}{4 \pi r_h}\]

\[\beta=\frac{1}{T}\]

GUP result

the \(\epsilon\):

\[2\sqrt{\lambda}=\int_{r_h}^{r_h+\epsilon}\frac{dr}{\sqrt{f}}\]

\[I(x)=\frac{f'(r_h)\epsilon^2 r_h^2}{2x^2(x^2+\frac{\beta^2 f'(r_h)\epsilon}{4 \lambda})^2}\]

where \(x=\frac{\beta \omega}{2}\)

Entropy

\[S_{AH}=\frac{\beta_{AH}^3}{12 \pi \lambda^3}\int_0^\infty dx \frac{x^4}{\sinh ^2 x} I(x)_{AH}\]

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