How to calculate the Hawking temperature of a "constant curvature black hole" by the Euclidean method?

Question

The constant curvature black hole (Bañados black hole) in $AdS_{d+1}$ is

$${\small ds^2=\frac{H^2}{\rho_+^2}(\rho^2-\rho_+^2)\left(-(1-H^2r^2)dt^2+\frac{dr^2}{1-H^2r^2}+r^2d\Omega_{d-2}^2\right)+\frac{d\rho^2}{\rho^2-\rho_+^2}+\rho^2d\phi^2.}$$

The AdS boundary is $dS_{d-1}\times S^1$. More information about this geometry can be found in [1,2]. How to calculate its temperature at the horizon $\rho=\rho_+$ by the Euclidean method? We can see that $g_{tt}$ depends on both coordinates $\rho$ and $r$, but its temperature should be constant.

The following example shows the Euclidean method to calculate the Hawking temperature of a black hole described by the metric

$$ds^2=-f(r)dt^2+\frac{dr^2}{g(r)}+r^2d\Omega_{2}^2,$$

where $f=f_0(r-r_h)+\cdots$, and $g=g_0(r-r_h)+\cdots$ near the horizon. Write the metric near the horizon as

$$ds^2=\kappa^2\rho^2d\tau^2+d\rho^2+\cdots.$$

To avoid conical singularity, the period of $\kappa\tau$ must be $2\pi$. The temperature is the inverse of the period of $\tau$:

$$T=\frac{\kappa}{2\pi}=\frac{\sqrt{f_0g_0}}{4\pi}.$$ ​

Another way to calculate the temperature is by the following formula for surface gravity:

$$\kappa^2=-\frac{1}{2}(\nabla^a\xi^b)(\nabla_a\xi_b),$$

where $\xi^a=(\partial_t)^a$. For the constant curvature black hole described above, we have

$$\kappa|_{\rho=\rho_+}=\kappa|_{r=1/H}=H.$$

So $T=H/2\pi.$ But I want to know whether the Euclidean method still works for the constant curvature black hole.

[1] M. Bañados, Constant Curvature Black Holes, gr-qc/9703040.
[2] D. Marolf, M. Rangamani, and M.V. Raamsdonk, Holographic models of de Sitter QFTs, arXiv:1007.3996.