  We consider solutions to the linear wave equation $\Box_g\phi=0$ on a
non-extremal maximally extended Schwarzschild-de Sitter spacetime arising from
arbitrary smooth initial data prescribed on an arbitrary Cauchy hypersurface.
(In particular, no symmetry is assumed on initial data, and the support of the
solutions may contain the sphere of bifurcation of the black/white hole
horizons and the cosmological horizons.) We prove that in the region bounded by
a set of black/white hole horizons and cosmological horizons, solutions $\phi$
converge pointwise to a constant faster than any given polynomial rate, where
the decay is measured with respect to natural future-directed advanced and
retarded time coordinates. We also give such uniform decay bounds for the
energy associated to the Killing field as well as for the energy measured by
local observers crossing the event horizon. The results in particular include
decay rates along the horizons themselves. Finally, we discuss the relation of
these results to previous heuristic analysis of Price and Brady et al.
