  We show that in any spacetime dimension $D\ge 4$, degenerate components of
the event horizon do not exist in static vacuum configurations with positive
cosmological constant. We also show that without a cosmological constant
asymptotically flat solutions cannot possess a degenerate horizon component.
Several independent proofs are presented. One proof follows easily from
differential geometry in the near-horizon limit, while others use
Bakry-\'Emery-Ricci bounds for static Einstein manifolds.
