<PERSON> temperature for an asymptotically $\text{AdS}_5$ black hole with a constant magnetic field
I'm trying to calculate the Hawking temperature for an asymptotically $\text{AdS}_5$ black hole with a constant magnetic field. Suppose that the metric and magnetic field of this background are given by
\begin{align} \text{d} s^2 &= \frac{L^2}{U(r)} \text{d} r^2 + \frac{1}{L^2} (-U(r) \text{d} t^2 + V(r) (\text{d} x^2 + \text{d} y^2) + W(r) \text{d} z^2) \tag{1},\ F &= B \text{d} x \wedge \text{d} y, \end{align}
such that $U(r), V(r), W(r) \rightarrow r^2$ as $r \rightarrow \infty$, and $U(r_h) = 0$, where $r_h$ is the horizon radius. If we expand at $r = r_h$ , the metric functions are approximately
\begin{align} U(r) &\simeq U_{1, h} (r - r_h),\ V(r) &\simeq V_{0, h},\ W(r) &\simeq W_{0, h}, \end{align}
where $U_{1, h}, \ V_{0, h}, \ W_{0, h}$, are expansion coefficients. 474
Plugging these expansions in the metric, and after a <PERSON> rotation, $\text{d} s^2$ becomes
$$\text{d} s^2 \simeq \frac{L^2}{U_{1, h} (r - r_h)} \text{d} r^2 + \frac{1}{L^2} (U_{1, h} (r - r_h) \text{d} t_{\text{E}}^2 + V_{0, h} (\text{d} x^2 + \text{d} y^2) + W_{0, h} \text{d} z^2).$$
Through the change of coordinates
\begin{align} \rho^2 &= \frac{4L^2 (r - r_h)}{U_{1, h}},\ \varphi &= \frac{U_{1, h} t_{\text{E}}}{2L^2}, \end{align}
the metric is
$$\text{d} s^2 = \text{d} \rho^2 + \rho^2 \text{d} \varphi^2 + \frac{1}{L^2} (V_{0, h} (\text{d} x^2 + \text{d} y^2) + W_{0, h} \text{d} z^2) \tag{2}.$$
In order to avoid a conic singularity as $\rho \rightarrow 0$, $\varphi$ has to have a period of $2 \pi$. On the other hand, when a <PERSON> rotation is made, the Euclidean time coordinate is identified with period $\beta = \frac{1}{T}$. Therefore the <PERSON> temperature is given by
$$T_{\text{H}} \stackrel{?}{=} \frac{U_{1, h}}{4 \pi L^2}.$$
However, the reason I doubt this result is because, as far as I understand, the metric has to be of the form
$$\text{d} s^2 = U(r) \text{d} t_{\text{E}}^2 + \frac{1}{U(r)} \text{d} r^2 + r^2 \text{d} \Omega_{n - 2}^2 \tag{3},$$
in order to arrive at the afore mentioned conclusion. Unless I'm mistaken, there's no change of coordinates in which (1) can be transformed into (3). 15