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We investigate the topological properties of the $SU(3)$ pure gauge theory byperforming numerical simulations at imaginary values of the $\theta$ parameter.By monitoring the dependence of various cumulants of the topological chargedistribution on the imaginary part of $\theta$ and exploiting analyticcontinuation, we determine the free energy density up to the sixth order orderin $\theta$, $f(\theta,T) = f(0,T) + {1\over 2} \chi(T) \theta^2 (1 + b_2(T)\theta^2 + b_4(T) \theta^4 + O(\theta^6))$. That permits us to achievedeterminations with improved accuracy, in particular for the higher orderterms, with control over the continuum and the infinite volume extrapolations.We obtain $b_2=-0.0216(15)$ and $|b_4|\lesssim 4\times 10^{-4}$.

$\theta$ dependence in $SU(3)$ Yang-Mills theory from analytic continuation