  Among the Scalar-Tensor modified theories of gravity, DHOST models could play
a special role for dark energy while being consistent with current
observations, notably those constraining the speed of gravitational waves.
Schwarzschild-de Sitter black holes were shown to be exact solutions of a
particular subclass of quadratic DHOST theories, while carrying a nontrivial
scalar profile that linearly evolves in time and hence potentially providing
exciting new phenomenological windows to explore this model. We investigate the
physical perturbations about such black holes and find that the odd-parity
tensor perturbations behave in a way indistinguishable to GR. On the other
hand, the effective metric for the (even-parity) scalar perturbations is
singular, indicating that those exact black hole solutions are infinitely
strongly coupled and cannot be trusted within the regime of validity of the
DHOST effective field theory. We show how this strong coupling result is
generalizable to a whole class of solutions with arbitrary manifolds both for
DHOST and Horndeski.
