= Solution
For the <static patch of de Sitter spacetime>, $r_*=\int dr/F=(R/2)\log|(R+r)/(R-r)|$. With $v=t+r_*$ or $u=t-r_*$, respectively,
$$
ds^2=-F\,dv^2+2\,dv\,dr+r^2d\Omega_2^2,\qquad ds^2=-F\,du^2-2\,du\,dr+r^2d\Omega_2^2.
$$
Both radial blocks have determinant $-1$ and smooth coefficients at $r=R$. The normal has norm $g^{rr}=F$, so that surface is null. There $k=\partial_v$ has covector $dr$, or $k=\partial_u$ has covector $-dr$, making it normal as well as tangent: it is a <Killing horizon>.
For the <signed surface gravity of a de Sitter horizon>, since $\partial_rk^2=-F'(R)=2/R$, part a gives $\kappa=-1/R$ in the ingoing chart on the past branch, and $\kappa=+1/R$ in the outgoing chart on the future branch, with $k$ future-directed in the static patch. \b[The physical magnitude is $\boxed{|\kappa|=1/R}$], normalized by central <proper time>. A negative signed $F'(R)/2$ must not become a negative temperature.
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