  We investigate black hole solutions with time-dependent (scalar) hair in
scalar-tensor theories. Known exact solutions exist for such theories at the
background level, where the metric takes on a standard GR form (e.g.
Schwarzschild-de Sitter), but these solutions are generically plagued by
instabilities. Recently, a new such solution was identified in
arXiv:2310.11919, in which the time-dependent scalar background profile is
qualitatively different from previous known exact solutions - specifically, the
canonical kinetic term for the background scalar $X$ is not constant in this
solution. We investigate the stability of this new solution by analysing odd
parity perturbations, identifying a bound placed by stability and the resulting
surviving parameter space. We extract the quasinormal mode spectrum predicted
by the theory, finding a generic positive shift of quasinormal mode frequencies
and damping times compared to GR. We forecast constraints on these shifts (and
the single effective parameter $\hat\beta$ controlling them) from current and
future gravitational wave experiments, finding constraints at up to the ${\cal
O}(10^{-2})$ and ${\cal O}(10^{-6})$ level for LVK and LISA/TianQin,
respectively. All calculations performed in this paper are reproducible via a
companion Mathematica notebook.
