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In this paper we investigate, by means of numerical lattice simulations, thetopological properties of the trace deformed $SU(3)$ Yang-Mills theory definedon $S_1\times\mathbb{R}^3$. More precisely, we evaluate the topologicalsusceptibility and the $b_2$ coefficient (related to the fourth cumulant of thetopological charge distribution) of this theory for different values of thelattice spacing and of the compactification radius. In all the cases we findresults in good agreement with the corresponding ones of the standard $SU(3)$Yang-Mills theory on $\mathbb{R}^4$.

$\theta$ dependence in trace deformed $SU(3)$ Yang-Mills theory: a lattice study