In this talk, we will address various aspects of hairy black holes that are solutions of four-dimensional gravity in the presence of a dilatonic scalar field and an Abelian gauge field. In particular, we will study their thermodynamics as a consequence of a well-posed variational principle. We find that for a slow fall-off of the scalar fields, they introduce a non-integrable term in the variation of the mass, that make the first law of black hole thermodynamics to be satisfied. The appearance of a non-integrable term is solved by proposing boundary conditions that arbitrarily relates the leading and subleading terms of the scalar field fall-off.

In a second part of the talk, we give a first attempt to connect thermodynamic black holes with astrophysical ones, where the presence of a non-integrable term will be crucial. We propose a way to connect two a priori distinct aspects of black hole physics: their thermodynamics, and their description as point particles, which is an essential starting point in the post-Newtonian approach to their dynamics. We will find that, when reducing a black hole to a point particle endowed with its specific effective mass, one in fact describes a black hole satisfying the first law of thermodynamics such that its global charges, and hence its entropy,
remain constant.