  In order to derive equation of state for the pure SU(3) Yang-Mills fields
from first principles, it is proposed to generalize the effective potential
approach for composite operators to non-zero temperatures. It is essentially
non-perturbative by construction, since it assumes the summation of an infinite
number of the corresponding contributions. There is no dependence on the
coupling constant, only a dependence on the mass gap, which is responsible for
the large-scale structure of the QCD ground state. The equation of state
generalizes the Bag constant at non-zero temperatures, while its nontrivial
Yang-Mills part has been approximated by the generalization of the free gluon
propagator to non-zero temperatures, as a first necessary step. Even in this
case we were able to show explicitly that the pressure may almost continuously
change its regime at $T^* = 266.5 MeV$.All the other thermodynamical quantities
such as energy density, entropy, etc. are to be understood to have drastic
changes in their regimes in the close vicinity of $T^*$. All this is in
qualitative and quantitative agreement with thermal lattice QCD results for the
pure Yang-Mills fields. We have firmly established the behavior of all the
thermodynamical quantities in the region of low temperatures, where thermal
lattice QCD calculations suffer from big uncertainties.
