Surface gravity of an asymptotically $\text{AdS}_5$ black hole with a constant magnetic field
A week ago, I posted this question about calculating the Hawking temperature for the following metric with <PERSON> field
\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 as $r \rightarrow \infty$, $U(r), \ V(r), \ W(r) = r^2$ and $U(r_h) = 0$, where $r_h$ is the horizon radius.
Through the Euclidean time formalism, in said post I calculated the <PERSON> temperature to be
$$T_{\text{H}} = \frac{U'(r_h)}{4 \pi L^2}.$$
I want to verify that I get the same result using the black hole's surface gravity. However, there's some things that make me doubtful of whether my analysis is correct, which is the following.
Since the metric doesn't depend on $t$, then $\xi^\mu \partial_\mu = \partial_t$ is a Killing vector field. Firstly,
$$\lim_{r \rightarrow r_h} g_{\mu \nu} \xi^\mu \xi^\nu = 0,$$
which means that $\xi^\mu \partial_\mu$ is null on the horizon. Secondly
$$\lim_{r \rightarrow r_h} g_{\mu \nu} \xi^\nu = 0 \tag{2},$$
so $\xi^\mu \partial_\mu$ is normal to the horizon. Thus $r = r_h$ is a Killing horizon. Also, $\xi^\mu \partial_\mu$ is hypersurface orthogonal since $\xi_{[\mu} \nabla_\nu \xi_{\rho]} = 0.$
With this in mind, we know that the surface gravity is given by
$$\kappa = \lim_{r \rightarrow r_h} \sqrt{-\frac{1}{2} (\nabla^\mu \xi^\nu) (\nabla_\mu \xi_\nu)},$$
and is constant along the horizon. 474
Through some calculations, $\kappa$ turns out to be
$$\kappa = \frac{U'(r_h)}{2L^2}.$$
Therefore, the Hawking temperature is (in natural units $c = \hbar = k_{\text{B}} = 1$)
$$T_{\text{H}} = \frac{\kappa}{2 \pi} = \frac{U'(r_h)}{4 \pi L^2} \tag{3},$$
which is the same result I got in my post from a week ago. However, here are the things that give me some trouble: I don't know if equation (2) really means that $\xi^\mu \partial_\mu$ is normal to the horizon. What I want to express in that equation, is the fact that the "inner product" between the metric and the Killing vector field as $r \rightarrow r_h$ is 0, yet I'm not sure if I'm interpreting said equation correctly.
The other thing that brings me some confusion is that, as far as I'm aware, these results are valid in asymptotically flat spacetimes, which mine is not. In particular, these results are the ones an observer far away from the black hole would measure. In contrast, I don't know if that's the case here since the metric is asymptotically $\text{AdS}5$. For example, in this post it is described that we can fix the Killing vector field by requiring $\xi^\mu \xi\mu|_\infty = -1$ which cannot be done in this case. Does this mean that I cannot apply the previous results in order to conclude that (3) is the <PERSON> temperature?
Thank you for your time. 474