= Solution
In the <ingoing Eddington-Finkelstein coordinates>, the stationary <Killing vector field> is $k=\partial_v$ with norm $k^2=-f$. The normal vector to a level set of $r$ is
$$
\nabla^ar=(\partial_v)^a+f(\partial_r)^a.
$$
At $r=r_+$ it equals $k^a$, which is null there. Thus this null hypersurface is a <Killing horizon> of $k$. Its <surface gravity> is fixed by $\nabla_a(k^2)=-2\kappa k_a$ on the horizon. Since $k_a=(dr)_a$ there and $\nabla_a(k^2)=-f'(r_+)(dr)_a$,
$$
\boxed{\kappa=\frac12f'(r_+)=\frac{r_+^2-r_-^2}{r_+^3}.}
$$
The positive sign follows from the future outer-horizon generator, normalized by the static time at infinity. It vanishes as the two radii approach the extremal limit.
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