  It has been known that in the presence of a scalar hair there would be a
distinct additional contribution to the first law of black hole thermodynamics.
While it has been checked in many examples, a deeper understanding of this
issue is necessary. The thermodynamics of AdS black holes in Einstein-scalar
gravity is studied by using the standard holographic renormalization procedure
and the variation of the Hamiltonian via the Wald formalism. It is found that
the first law requires a modification by including an additional term that has
a particular form $\sim\left<O\right>\delta \phi_s$, with $\phi_s$ and
$\left<O\right>$ a new pair of thermodynamic conjugate variables. $\phi_s$ is
the leading source term of the asymptotic fall-off of the scalar field near the
AdS boundary, and $\left<O\right>$ is precisely the response of the dual scalar
operator from the holographic point of view. Some hairy black holes are
constructed explicitly to check the first law of thermodynamics as well as the
thermodynamic relations.
