Heat capacity of the nuclear-spin system (NSS) in GaAs-based microstructures has been shown to be much greater than expected from dipolar coupling between nuclei, thus limiting the efficiency of NSS cooling by adiabatic demagnetization. It was suggested that quadrupole interaction induced by some small residual strain could provide this additional reservoir for the heat storage. We check and validate this hypothesis by combining nuclear-spin relaxation measurements with adiabatic remagnetization and nuclear magnetic resonance experiments, using electron spin-noise spectroscopy as a unique tool for detection of nuclear magnetization. Our results confirm and quantify the role of the quadrupole splitting in the heat storage within NSS and provide additional insight into the fundamental, but still actively debated relation between a mechanical strain and the resulting electric field gradients in GaAs.