  We investigate the dependence of four-dimensional SU(N) gauge theories on the
topological theta term at finite temperature and, in particular, across the
deconfinement transition. For this purpose, we exploit the lattice formulation
of the theory and present numerical results for the expansion of the free
energy up to O(theta^6), for N=3 and N=6.
  Our numerical analysis shows that the theta dependence of 4D SU(N) gauge
theory experiences a drastic change across the deconfinement transition: the
low-temperature phase is characterized by a large-N scaling with theta/N as
relevant variable, while in the high-temperature phase the scaling variable is
just theta and the free energy is essentially determined by the instanton-gas
approximation. The crossover between the two different behaviours gets sharper
with increasing N, suggesting that the instanton-gas regime sets in just above
Tc at large N.
