  In this paper we study, by means of numerical simulations, the behaviour of
the scalar glueball mass and the ground state of the torelon for trace deformed
Yang-Mills theory defined on $ \mathbb{R}^3\times S^1$, in which center
symmetry is recovered even at small compactification radii. We find, by
investigating both $SU(3)$ and $SU(4)$ pure gauge theories, that the glueball
mass computed in the deformed theory, when center symmetry is recovered, is
compatible with its value at zero temperature and does not show any significant
dependence on the compactification radius; moreover, we establish a connection
between the deformation parameter and an effective compactification size, which
works well at least for small deformations. In addition, we observe that the
ground state of the torelon which winds around the small traced deformed circle
with size $l$ acquires a pleateau for large values of the strength $h$, with
values which are compatible with a $1/l$ behavior but, on the other hand, are
still not in complete agreement with the asymptotic semiclassical large-$N$
predictions.
