  We present a proposal for calculating the running of the coupling constant of
the $\mathrm{SU}(3)$ pure-gauge theory, which combines the Twisted Gradient
Flow (TGF) renormalization scheme with Parallel Tempering on Boundary
Conditions (PTBC). The TGF is a gradient flow-based renormalization scheme
formulated in an asymmetric lattice with twisted boundary conditions. Combined
with step scaling, it has been successfully used to calculate the
$\mathrm{SU}(3)$ $\Lambda$ parameter. As with all gradient flow-based schemes,
the coupling constant is highly correlated with the topological charge and
affected by topology freezing, an issue addressed by projecting the
determination of the coupling onto the zero topological sector. As an
alternative to the zero-charge projection, we combine TGF with PTBC, by
replicating multiple copies of the same lattice, interpolating between periodic
and open boundary conditions in a parallel-tempered manner. We present a first
exploration of these ideas by analyzing specific ensembles of $\mathrm{SU}(3)$
lattices with and without PTBC.

High Energy Physics - Lattice