The possibility to perform high-sensitivity X-ray phase-contrast imaging with laboratory grating-based phase-contrast computed tomography (gbPC-CT) setups is of great interest for a broad range of high-resolution biomedical applications. However, achieving high sensitivity with laboratory gbPC-CT setups still poses a challenge because several factors such as the reduced flux, the polychromaticity of the spectrum, and the limited coherence of the X-ray source reduce the performance of laboratory gbPC-CT in comparison to gbPC-CT at synchrotron facilities. In this work, we present our laboratory X-ray Talbot-Lau interferometry setup operating at 40 kVp and describe how we achieve the high sensitivity yet unrivalled by any other laboratory X-ray phase-contrast technique. We provide the angular sensitivity expressed via the minimum resolvable refraction angle both in theory and experiment, and compare our data with other differential phase-contrast setups. Furthermore, we show that the good stability of our high-sensitivity setup allows for tomographic scans, by which even the electron density can be retrieved quantitatively as has been demonstrated in several preclinical studies.