  According to static patch holography, de Sitter space admits a unitary
quantum description in terms of a dual theory living on the stretched horizon,
that is a timelike surface close to the cosmological horizon. In this
manuscript, we compute several holographic complexity conjectures in a periodic
extension of the Schwarzschild-de Sitter black hole. We consider multiple
configurations of the stretched horizons to which geometric objects are
anchored. The holographic complexity proposals admit a hyperfast growth when
the gravitational observables only lie in the cosmological patch, except for a
class of complexity=anything observables that admit a linear growth. All the
complexity conjectures present a linear increase when restricted to the black
hole patch, similar to the AdS case. When both the black hole and the
cosmological regions are probed, codimension-zero proposals are
time-independent, while codimension-one proposals can have non-trivial
evolution with linear increase at late times. As a byproduct of our analysis,
we find that codimension-one spacelike surfaces are highly constrained in
Schwarzschild-de Sitter space. Therefore, different locations of the stretched
horizon give rise to different behaviours of the complexity conjectures.
