  We study the topological characterization of the energy gaps in general
two-dimensional quasiperiodic systems consisting of multiple periodicities,
represented by twisted two-dimensional materials. We show that every single gap
is uniquely characterized by a set of integers, which quantize the area of the
momentum space in units of multiple Brillouin zones defined in the redundant
periodicities. These integers can be expressed as the second Chern numbers, by
considering an adiabatic charge pumping under a relative slide of different
periodicities, and using a formal relationship to the four-dimensional quantum
Hall effect. The integers are independent of commensurability of the multiple
periods, and invariant under arbitrary continuous deformations such as a
relative rotation of twisted periodicities.
