gauge theory time has to be $\tau = t \sqrt{ f(z_0) } $. Since the period is the inverse of the gauge theory temperature: $ \tau \sim \tau + 1/T$ and the period of the black hole coordinate $t$ must be $\pi z_h$ to avoid a conical singularity at the horizon, one finds:

$$T = \frac{1}{ \pi z_h \sqrt{ f (z_0) }}  = \frac{ 1}{ \pi z_h \sqrt{ 1 - \frac{z_0^4 }{z_h^ 4 }  \,}}\,.
  \label{temp}$$

The action densities for the black hole AdS and thermal AdS in the soft wall model are calculated in ref. [@Herzog:2006ra]. The results of this article can be adapted to the model considered here, where there is an UV cutoff, by replacing the minimum value of the coordinate $z$ that in ref. [@Herzog:2006ra] is just an UV regulator $ z= \epsilon $ by the (inverse of the) UV energy scale: $z = z_0$. Using also the relation between the horizon position and the temperature in eq. (\[temp\]) one gets:

$$\begin{aligned}
 V_{th\, AdS} &=& \frac{4  R^ 3  }{g_5^2} \,\frac{1}{T}  \, \int_{z_0}^\infty dz \frac{ e^{-k^ 2 z^ 2}}{z^ 5} 
 \label{densities1} \\
 V_{BH \, AdS}  &=&   \frac{4 R^ 3  }{g_5^2} \, \frac{1}{T \sqrt{f(z_0) }} \int_{z_0}^{z_h} dz \frac{ 