  While regular flat bands are good for enhancing the density of states and
hence the gap, they are detrimental to the superfluid weight. We show that the
predicted nontrivial topology of the two lowest flat bands of twisted bilayer
graphene plays an important role in the enhancement of the superfluid weight
and hence of superconductivity. We derive the superfluid weight (phase
stiffness) of the TBLG superconducting flat bands with a uniform pairing, and
show that it can be expressed as an integral of the Fubini-Study metric of the
flat bands. This mirrors results already obtained for nonzero Chern number
bands even though the TBLG flat bands have zero Chern number. We further show
the metric integral is lower bounded by the topological $C_{2z}T$ Wilson loop
winding number of the TBLG flat bands, which renders the superfluid weight has
a topological lower bound proportional to the pairing gap. In contrast, trivial
flat bands have a zero superfluid weight. The superfluid weight is crucial in
determining the BKT transition temperature of the superconductor. Based on the
transition temperature measured in TBLG experiments, we estimate the
topological contribution of the superfluid weight in TBLG.
