Experiments with bosonic atoms in optical superlattices allow for the interesting possibility to study the adiabatic quantized pumping of bosonic atoms in the presence of interactions. We investigate this exotic phenomenon for bound bosonic pairs in the paradigmatic Su-Schrieffer-Heeger model where the ground state exhibits topological properties due to dimerized hoppings. We show that at unit filling there exist crossovers and phase transitions to bond-order phases of paired bosons known as the pair-bond-order phase for attractive interactions. The pair-bond-order phase is found to exhibit effective topological properties such as the presence of polarized paired edge states. This is further analyzed by studying the emergence and breakdown of the Thouless charge pumping of bound bosonic pairs by a parametric extension to the famous Rice-Mele model. Finally, we discuss how the pumping of paired bosons or different regimes of breakdown of charge pumping can be probed by state-of-the-art experiments, for example, with repulsively bound bosons.