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We investigate the topological properties of unquenched $QCD$ on the basis ofnumerical results of simulations at fixed topological charge, recently reportedby Borsanyi et al.. We demonstrate that their results for the mean value of thechiral condensate at fixed topological charge are inconsistent with theanalytical prediction of the large volume expansion around the saddle point,and argue that the most plausible explanation for the failure of the saddlepoint expansion is a vacuum energy density $\theta$-independent at hightemperatures, but surprisingly not too high $(T\sim 2T_c)$, a result whichwould imply a vanishing topological susceptibility, and the absence of allphysical effects of the $U(1)$ axial anomaly at these temperatures. We alsoshow that under a general assumption concerning the high temperature phase of$QCD$, where the $SU(N_f)_A$ symmetry is restored, the analytical predictionfor the chiral condensate at fixed topological charge is in very good agreementwith the numerical results of Borsanyi et al., all effects of the axial anomalyshould disappear, the topological susceptibility and all the$\theta$-derivatives of the vacuum energy density vanish and the theory becomes$\theta$-independent at any $T>T_c$ in the infinite volume limit.

Topology in the SU(Nf) chiral symmetry restored phase of unquenched QCD and axion cosmology