Hairy black holes (BHs) have macroscopic degrees of freedom which are not associated with a Gauss law. As such, these degrees of freedom are not manifest as quasi-local quantities computed at the horizon. This suggests conceiving hairy BHs as an interacting system with two components: a "bald" horizon coupled to a "hairy" environment. Based on this idea we suggest an effective model for hairy BHs -typically described by numerical solutions -that allows computing analytically thermodynamic and other quantities of the hairy BH in terms of a fiducial bald BH. The effective model is universal in the sense that it is only sensitive to the fiducial BH, but not to the details of the hairy BH. Consequently, it is only valid in the vicinity of the fiducial BH limit. We discuss, quantitatively, the accuracy of the effective model for asymptotically flat BHs with synchronised hair, both in D = 4 (including self-interactions) and D = 5 spacetime dimensions. We also discuss the applicability of the model to synchronised BHs in D = 5 asymptotically AdS and static D = 4 coloured BHs, exhibiting its limitations.arXiv:1803.09089v1 [gr-qc] ties up nicely with the microscopic picture, and the view that the horizon contains all relevant BH information.The discovery of "hairy" BHs in a variety of models (see e.g. [8-10] for reviews) has overshadowed this conceptually simple picture. These BHs have extra macroscopic degrees of freedom not associated to a Gauss law. Therefore they do not seem to be associated to any quasi-local conserved quantity computable at the horizon level. This raises interesting questions, on how the microscopic description of the BH captures these extra macroscopic degrees of freedom, but it also suggests an effective model for obtaining an (in general) analytic approximation for physical and thermodynamical quantities of the hairy BHs associated to the horizon [11].The basic idea of the effective model sketched in [11] (and suggested by the numerical evolutions in [12]), therein called quasi-Kerr horizon model, is that due to the absence of further local charges at the horizon, the horizon of the hairy BH is well approximated by the horizon of a fiducial bald BH but with different parameters. In a sense, the hairy BH can be conceived as a coupled system of a bald horizon with an external "hair" environment. Naturally, the system is interacting and the non-linearities of the underlying gravity-matter system introduce a non-trivial deformation of the "bald" horizon. But in the feeble hair regime, when only a small percentage of the overall spacetime energy is contained in the matter field, these non-linearities are expected to be small, and the horizon should still behave as that of the bald fiducial BH, but with shifted parameters to take into account the mass and angular momenta that is no longer inside the horizon but rather in the matter environment. One could expect such simple model to yield errors in the thermodynamics quantities of the order of the deviation from the fiducial bald BH. The findings in [1...