  Euclidean strong coupling expansion of the partition function is applied to
lattice Yang-Mills theory at finite temperature, i.e. for lattices with a
compactified temporal direction. The expansions have a finite radius of
convergence and thus are valid only for $\beta<\beta_c$, where $\beta_c$
denotes the nearest singularity of the free energy on the real axis. The
accessible temperature range is thus the confined regime up to the
deconfinement transition. We have calculated the first few orders of these
expansions of the free energy density as well as the screening masses for the
gauge groups SU(2) and SU(3). The resulting free energy series can be summed up
and corresponds to a glueball gas of the lowest mass glueballs up to the
calculated order. Our result can be used to fix the lower integration constant
for Monte Carlo calculations of the thermodynamic pressure via the integral
method, and shows from first principles that in the confined phase this
constant is indeed exponentially small. Similarly, our results also explain the
weak temperature dependence of glueball screening masses below $T_c$, as
observed in Monte Carlo simulations. Possibilities and difficulties in
extracting $\beta_c$ from the series are discussed.
