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

We study the dependence of 4D SU(N) gauge theories on the topological θ term at finite temperature T. We exploit the lattice formulation of the theory, presenting numerical results for the expansion of the free energy up to O(θ6), for N=3 and N=6. Our analysis shows that the θ dependence drastically changes across the deconfinement transition: the low-T phase is characterized by a large-N scaling with θ/N as relevant variable, while in the high-T phase the scaling variable is just θ and the free energy is essentially determined by the instanton-gas approximation.