In quantum materials research, thermodynamic measurements are a key tool for probing the underlying physics of materials. The highest quality samples are often only available as microcrystals of typical mass < 100 μg. Heat capacity measurements on such small samples are beyond the scope of conventional set-ups but have been demonstrated in membrane-based calorimeters. This motivates the development of a nanocalorimeter to measure microcrystals of order nJ/K heat capacity, at ultra low temperature and high magnetic field. A key challenge of such a nanocalorimeter is the need for effectively primary thermometry, to diagnose any decoupling of the sample thermometer from the environment. This aim is realised by implementing the technique of Coulomb blockade thermometry (CBT) onto a membrane-based calorimeter. The proof of principle of this novel nanocalorimeter is shown by the successful fabrication and measurement of sensitive CBT structures onto a 200 nm thin silicon nitride membrane. The nanocalorimeter accuracy is tested by measuring the well-studied material Sr₃Ru₂O₇. We successfully measure a Sr₃Ru₂O₇ microcrystal with heat capacity of ~3 nJ/K, and find remarkably good agreement with literature measurements on samples three orders of magnitude larger. The newly developed nanocalorimeter is then applied to measure the topical material CeRh₂As₂, a heavy-fermion superconductor with a rare field induced transition. The CeRh₂As₂ phase diagram is well studied, however reported phase transitions have significant widths of > 10% of the transition temperatures. We measure a CeRh₂As₂ microcrystal with the intention of observing reduced disorder broadening. Indeed we consistently observe sharper and more distinct transitions than comparable specific heat measurements on larger samples [4]. For the superconducting transition we determine Tc = 0.36 K, which is higher than previously reported for a larger crystal of the same growth generation. The phase diagram we present shows the intersection of putative QDW transition line with the SC dome at 6 T. Furthermore, low temperature field dependent specific heat and analysis of ΔC/C hint at the possibility of three distinct regimes within the SC dome. With further improvements to the next generation of nanocalorimeter, these details will be investigated further.