  We consider scalegenesis, spontaneous scale symmetry breaking, by the
scalar-bilinear condensation in $SU(N)$ scalar gauge theory. In an effective
field theory approach to the scalar-bilinear condensation at finite
temperature, we include the Polyakov loop to take into account the confinement
effect. The theory with $N=3,4,5$ and $6$ is investigated, and we find that in
all these cases the scale phase transition is a first-order phase transition.
We also calculate the latent heat at and slightly below the critical
temperature. Comparing the results with those obtained without the Polyakov
loop effect, we find that the Polyakov effect can considerably increase the
latent heat in some cases, which would mean a large increase in the energy
density of the gravitational waves background, if it were produced by the scale
phase transition.
