= Solution
In advanced coordinates, $K=\partial_v$ is a <Killing vector field> and $K^2=-f$. The normal covector to $r=2m$ is $dr$; raising it gives $g^{ar}\partial_a=\partial_v+f\partial_r$, which equals $K$ on the horizon. Thus $r=2m$ is a <null hypersurface> normal to $K$, and hence a <Killing horizon>.
The <surface gravity> is defined by $\nabla_KK=\kappa K$ on this horizon, equivalently $\nabla_a(K^2)=-2\kappa K_a$. Here $K_a=(-f,1,0,0)$, so the radial component gives $\kappa=f'(2m)/2$. Because $F(2m)=0$,
$$
\boxed{\kappa=\frac12\frac{F'(2m)}{\sqrt{H(2m)}}=\frac1{4m\cosh\alpha}.}
$$
It is positive for all finite real $\alpha$ and $m>0$; the stationary <Killing vector field> retains its asymptotic normalization.
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