= Solution
The normal to $r=r_+$ is $dr$, whose squared norm is $g^{rr}=f(r_+)=0$, so the surface is a <null hypersurface>. In ingoing coordinates $K=\partial_v$, and on the horizon
$$
K_a=g_{av}=(0,1,0,\ldots)=\nabla_ar.
$$
Thus $K$ is both tangent and normal there, making the surface a <Killing horizon>. For a static metric of this form, the <surface gravity> is $\kappa=f'(r_+)/2$. With $n=d-3$,
$$
\boxed{\kappa=\frac{d-3}{2r_+}
\left[1-\left(\frac{r_-}{r_+}\right)^{d-3}\right]}.
$$
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