  Recent experiments with ultracold quantum gases have successfully realized
integer-quantized topological charge pumping in optical lattices. Motivated by
this progress, we study the effects of static disorder on topological Thouless
charge pumping. We focus on the half-filled Rice-Mele model of free spinless
fermions and consider random diagonal disorder. In the instantaneous basis, we
compute the polarization, the entanglement spectrum, and the local Chern
marker. As a first main result, we conclude that the space-integrated local
Chern marker is best suited for a quantitative determination of topological
transitions in a disordered system. In the time-dependent simulations, we use
the time-integrated current to obtain the pumped charge in slowly periodically
driven systems. As a second main result, we observe and characterize a
disorder-driven breakdown of the quantized charge pump. There is an excellent
agreement between the static and the time-dependent ways of computing the
pumped charge. The topological transition occurs well in the regime where all
states are localized on the given system sizes and is therefore not tied to a
delocalization-localization transition of Hamiltonian eigenstates. For
individual disorder realizations, the breakdown of the quantized pumping occurs
for parameters where the spectral bulk gap inherited from the band gap of the
clean system closes, leading to a globally gapless spectrum. As a third main
result and with respect to the analysis of finite-size systems, we show that
the disorder average of the bulk gap severely overestimates the stability of
quantized pumping. A much better estimate is the typical value of the
distribution of energy gaps, also called mode of the distribution.
