  We construct concrete counterterms of the Balasubramanian-Kraus type for
Einstein-scalar theories with designer gravity boundary conditions in
AdS$_{4}$, so that the total action is finite on-shell and satisfy a well
defined variational principle for an arbitrary scalar field potential. We focus
on scalar fields with the conformal mass, $m^{2}=-2l^{-2}$, and show that the
holographic mass matches the Hamiltonian mass for any boundary conditions. We
compute the trace anomaly of the dual field theory in the generic case, as well
as when there exist logarithmic branches of non-linear origin. As expected, the
anomaly vanishes for the boundary conditions that are AdS invariant. When the
anomaly does not vanish, the dual stress tensor describes a thermal gas with an
equation of state related to the boundary conditions of the scalar field. When
the anomaly vanishes, we recover the dual theory of a massless thermal gas. As
an application of the formalism, we consider a general family of exact hairy
black hole solutions that, for some particular values of the parameters in the
moduli potential, contains solutions of four-dimensional gauged $\mathcal{N}=8$
supergravity and its $\omega$-deformation. Using the AdS/CFT duality
dictionary, they correspond to triple trace deformations of the dual field
theory.
