  We investigate the thermodynamical behavior and the scaling symmetries of the
scalar dressed black brane (BB) solutions of a recently proposed, exactly
integrable Einstein-scalar gravity model [1], which also arises as
compactification of (p-1)-branes with a smeared charge. The extremal, zero
temperature, solution is a scalar soliton interpolating between a conformal
invariant AdS vacuum in the near-horizon region and a scale covariant metric
(generating hyperscaling violation on the boundary field theory)
asymptotically. We show explicitly that for the boundary field theory this
implies the emergence of an UV length scale (related to the size of the brane),
which decouples in the IR, where conformal invariance is restored. We also show
that at high temperatures the system undergoes a phase transition. Whereas at
small temperature the Schwarzschild-AdS BB is stable, above a critical
temperature the scale covariant, scalar-dressed BB solution, becomes
energetically preferred. We calculate the critical exponent z and the
hyperscaling violation parameter of the scalar-dressed phase. In particular we
show that the hyperscaling violation parameter is always negative. We also show
that the above features are not a peculiarity of the exact integrable model of
Ref.[1], but are a quite generic feature of Einstein-scalar and
Einstein-Maxwell-scalar gravity models for which the squared-mass of the scalar
field is positive and the potential vanishes exponentially as the scalar field
goes to minus infinity.
