  In our previous works the effective potential approach for composite
operators has been generalized to non-zero temperature in order to derive the
analytical equation of state for pure SU(3) Yang-Mills fields without quark
degrees of freedom. In the absence of external sources this is nothing but the
vacuum energy density. The key element of this derivation is the introduction
of a temperature dependence into the expression for the bag constant. The
non-perturbative part of the analytical equation of state does not depend on
the coupling constant, but instead introduces a dependence on the mass gap.
This is responsible for the large-scale dynamical structure of the QCD ground
state. The perturbative part of the analytical equation of state does depend on
the QCD fine-structure coupling constant $\alpha_s$. Here we develop the
analytical formalism, incorporating the perturbative part in a self-consistent
way. It makes it possible to calculate the PT contributions to the equation of
state in terms of the convergent series in integer powers of a small
$\alpha_s$. We also explicitly derive and numerically calculate the first
perturbative contribution of the $\alpha_s$-order to the non-perturbative part
of the equation of state derived and calculated previously. The analytical
equation of state or, equivalently, the gluon pressure is exponentially
suppressed at low temperatures, while at temperature $T=T_c = 266.5 \ \MeV$ it
has a maximum, if divided by $T^4/3$. It demonstrates a highly non-trivial
dependence on the mass gap and the temperature near to $T_c$ and up to
approximately $(3-4)T_c$. At very high temperatures its polynomial character is
confirmed, containing the terms proportional to $T^2$ and $T$ with a
non-analytical dependence on the mass gap.
