  A scalar field non-minimally coupled to certain geometric [or matter]
invariants which are sourced by [electro]vacuum black holes (BHs) may
spontaneously grow around the latter, due to a tachyonic instability. This
process is expected to lead to a new, dynamically preferred, equilibrium state:
a scalarised BH. The most studied geometric [matter] source term for such
spontaneous BH scalarisation is the Gauss-Bonnet quadratic curvature [Maxwell
invariant]. This phenomenon has been mostly analysed for asymptotically flat
spacetimes. Here we consider the impact of a positive cosmological constant,
which introduces a cosmological horizon. The cosmological constant does not
change the local conditions on the scalar coupling for a tachyonic instability
of the scalar-free BHs to emerge. But it leaves a significant imprint on the
possible new scalarised BHs. It is shown that no scalarised BH solutions exist,
under a smoothness assumption, if the scalar field is confined between the BH
and cosmological horizons. Admitting the scalar field can extend beyond the
cosmological horizon, we construct new scalarised BHs. These are asymptotically
de Sitter in the (matter) Einstein-Maxwell-scalar model, with only mild
difference with respect to their asymptotically flat counterparts. But in the
(geometric) extended-scalar-tensor-Gauss-Bonnet-scalar model, they have
necessarily non-standard asymptotics, as the tachyonic instability dominates in
the far field. This interpretation is supported by the analysis of a test
tachyon on a de Sitter background.
