  The no-hair theorem postulates that the only externally observable properties
of a black hole are its mass, its electric charge, and its angular momentum. In
scalar-tensor theories of gravity, a matter distribution around a black hole
can lead to the so called "spontaneous scalarisation" instability that triggers
the development of scalar hair. In the Brans-Dicke type theories, this effect
can be understood as a result of tachyonic effective mass of the scalar field.
Here we consider the instability in the generalised class of scalar-theories
that feature non-conformal, i.e. "disformal", couplings to matter. Such
theories have gained considerable interest in the recent years and have been
studied in a wide variety of systems, both cosmological and astrophysical. In
view of the prospects of gravitational wave astronomy, it is relevant to
explore the implications of the theories in the strong-gravity regime. In this
article, we concentrate on the spontaneous scalarisation of matter
configurations around Schwarzschild and Kerr black holes. We find that in the
more generic scalar-tensor theories, the instability of the Brans-Dicke theory
can be enhanced, suggesting violations of the no-hair theorem. On the other
hand, we find that, especially if the coupling is very strong, or if the
gradients in the matter distribution are negligible, the disformal coupling
tends to stabilise the system.
