  We investigate the role of topology in the lattice determination of the
renormalized strong coupling via the gradient flow. This is done adopting the
Parallel Tempering on Boundary Conditions to deal with the infamous topological
freezing problem that affects standard local updating algorithms. We show that,
even in the presence of severe topological freezing, both algorithms yield the
same strong coupling once projected onto a fixed topological sector. Moreover,
we show that using a non-projected definition of the coupling leads to the same
step-scaling sequence. This means that projecting the coupling onto a fixed
topological sector does not affect the determination of the
dynamically-generated scale of the theory $\Lambda$.
