  A general approach how to analytically describe and understand $SU(3)$
lattice thermodynamics in the whole temperature range $[0, \infty)$ is
formulated and used. It is based on the effective potential approach for
composite operators properly extended to non-zero temperature and density. This
makes it possible to introduce into this general formalism the mass gap, which
is responsible for the large-scale dynamical structure of the QCD ground state.
The mass gap dependent gluon plasma pressure adjusted by this approach to the
corresponding lattice data is shown to be a continuously growing function of
temperature being thus differentiable in every point of its domain. At the same
time, the entropy and energy densities have finite jump discontinuities at some
characteristic temperature $T_c = 266.5 \ \MeV$ with latent heat
$\epsilon_{LH}= 1.41$. This is a firm evidence of the first-order phase
transition in $SU(3)$ pure gluon plasma. The heat capacity has a $\delta$-type
singularity (an essential discontinuity) at $T_c$, so that the velocity of
sound squared becomes zero at this point. All the independent thermodynamic
quantities are exponentially suppressed below $T_c$ and rather slowly approach
their respective Stefan-Boltzmann limits at high temperatures. Those
thermodynamic quantities which are the ratios of their independent counterparts
such as conformity, conformality and the velocity of sound squared approach
their Stefan-Boltzmann limits rather rapidly and demonstrate a non-trivial
dependence on the temperature below $T_c$. We also calculate the trace anomaly
relation (the interaction measure) and closely related to it the gluon
condensate, which are especially sensitive to the non-perturbative effects. An
analytical description of the dynamical structure of $SU(3)$ gluon plasma is
given.
