  We prove that any analytic vacuum spacetime with a positive cosmological
constant in four and higher dimensions, that contains a static extremal Killing
horizon with a maximally symmetric compact cross-section, must be locally
isometric to either the extremal Schwarzschild de Sitter solution or its
near-horizon geometry (the Nariai solution). In four-dimensions, this implies
these solutions are the only analytic vacuum spacetimes that contain a static
extremal horizon with compact cross-sections (up to identifications). We also
consider the analogous uniqueness problem for the four-dimensional extremal
hyperbolic Schwarzschild anti-de Sitter solution and show that it reduces to a
spectral problem for the laplacian on compact hyperbolic surfaces, if a
cohomological obstruction to the uniqueness of infinitesimal transverse
deformations of the horizon is absent.
