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

Change of theta dependence in 4D SU(N) gauge theories across the deconfinement transition