gauge theory case (in a non compact space) the black hole is the stable solution for all temperatures[@Witten:1998zw]. So, the dual geometry is the black hole. For a non conformal gauge theory, as in the soft wall model case, the dual geometry has two different phases, as discussed in refs. [@Herzog:2006ra; @BallonBayona:2007vp]. For temperatures above a critical value $T_c$ the black hole is stable while for temperatures below $T_c$ the thermal AdS is stable. The so called Hawking Page transition between spaces was interpreted in [@Witten:1998zw] as a transition in the dual gauge theory from a deconfined ($T > T_c$) to a confined phase ($T < T_c$).

In order to compare the action integrals of the black hole AdS and the thermal AdS we must take into account the fact that the periodicity of the time coordinate is related to the temperature. In our model the gauge theory is at $ z= z_0$ where the transverse part of the metric of the black hole is: $$ds^ 2 = \frac{R^ 2}{z_0^ 2}    \Big(    f(z_0) dt^2 +  d\vec{x}\cdot d\vec{x} \Big) \,.$$ The mapping of the supergravity theory to a gauge theory in flat space must be performed with the rule that the 