I am trying to calculate the Hawking temperature of a Schwarzschild black hole in a spacetime which is asymptotically dS. Ignoring the 2-sphere, the metric is given by $ds^2=\left(1-\frac{2M}{r}-\frac{r^2}{L^2}\right)d\tau^2+\left(1-\frac{2M}{r}-\frac{r^2}{L^2}\right)^{-1}dr^2$
where $\tau=it$ the Euclidean time and $L^2=\frac{3}{\Lambda}$.
In asymptotically flat space ($\Lambda=0$), one has to require that $\tau$ be periodic in the inverse temperature $\beta$ in order to prevent a conical singularity at the event horizon, from which the temperature follows.
In asymptotically de Sitter spacetime however, there are 2 positive roots of $g_{\tau\tau}$: the event horizon of the black hole $r_h$, but also the cosmological horizon $r_c>r_h$. As in the flat case, we can deduce the period of $\tau$ that is needed to prevent a conical singularity at $r_h$, but then we're still left with a conical singularity at $r_c$. Similarly we could make $\tau$ periodic in a way such that the conical singularity at $r_c$ disappears.
However, we cannot make both conical singularities disappear! Then how can we derive the black hole's Hawking temperature in this case? Should I just ignore the singularity at the cosmological horizon? Or should I use different coordinate patches?