  Recent experimental findings have reported the presence of unconventional
charge orders in the enlarged ($2 \times 2$) unit-cell of kagome metals
AV$_3$Sb$_5$ (A=K,Rb,Cs) and hinted towards specific topological signatures.
Motivated by these discoveries, we investigate the types of topological phases
that can be realized in such kagome superlattices. In this context, we employ a
recently introduced statistical method capable of constructing topological
models for any generic lattice. By analyzing large data sets generated from
symmetry-guided distributions of randomized tight-binding parameters, and
labeled with the corresponding topological index, we extract physically
meaningful information. We illustrate the possible real-space manifestations of
charge and bond modulations and associated flux patterns for different
topological classes, and discuss their relation to present theoretical
predictions and experimental signatures for the AV$_3$Sb$_5$ family.
Simultaneously, we predict new higher-order topological phases that may be
realized by appropriately manipulating the currently known systems.

Strongly Correlated Electrons