  We study the thermodynamics of $n$-dimensional static asymptotically AdS
black holes in Einstein gravity coupled to a scalar field with a potential
admitting a stationary point with an AdS vacuum. Such black holes with
non-trivial scalar hair can exist provided that the mass-squared of the scalar
field is negative, and above the Breitenlohner-Freedman bound. We use the Wald
procedure to derive the first law of thermodynamics for these black holes,
showing how the scalar hair (or "charge") contributes non-trivially in the
expression. We show in general that a black hole mass can be deduced by
isolating an integrable contribution to the (non-integrable) variation of the
Hamiltonian arising in the Wald construction, and that this is consistent with
the mass calculated using the renormalised holographic stress tensor and also,
in those cases where it is defined, with the mass calculated using the
conformal method of Ashtekar, Magnon and Das. Similar arguments can also be
given for the smooth solitonic solutions in these theories. Neither the black
hole nor the soliton solutions can be constructed explicitly, and we carry out
a numerical analysis to demonstrate their existence and to provide approximate
checks on some of our thermodynamic results.

High Energy Physics - Theory;General Relativity and Quantum Cosmology