  Hairy black-holes are a unique prediction of certain theories that extend
General Relativity (GR) with a scalar field. The presence of scalar hair is
reflected non-trivially in the entropy of the black hole along with any
topological coupling that may be present in the action. Demanding that a system
of two merging black holes obeys the global second law of thermodynamics
imposes a bound on this topological coupling coefficient. In this work we study
how this bound is pushed from its GR value by the presence of scalar hair by
considering estimates of binary black-hole merger parameters through inference
studies of mock gravitational-wave (GW) events. We find that the scalar charge
may produce a statistically significant deviation of the change in entropy over
the GR prediction. However, we also find the latter is susceptible to biases
arising out of GW inferences which ends up being two orders of magnitude
larger, therefore overwhelming any change induced by the scalar hair.
