  We explore the use of the recently defined scalar charge which satisfies a
Gauss law in stationary spacetimes, in the context of theories with a scalar
potential. We find new conditions that this potential has to satisfy in order
to allow for static, asymptotically-flat black-hole solutions with regular
horizons and non-trivial scalar field. These conditions are equivalent to some
of the known ``no-hair'' theorems (such as Bekenstein's). We study the extended
thermodynamics of these systems, deriving a first law and a Smarr formula. As
an example, we study the Anabal\'on-Oliva hairy black hole
