In a recent work we explored the backreaction of a self-gravitating system of elementary neutral fermions at finite temperature in asymptotically AdS space, known as the "holographic neutron star". Such study was carried out by solving the Tolman-Oppenhemier-Volkoff equations under a perfect fluid assumption for the fermionic equation of state, accounting for both relativistic and finite temperature effects. Novel "dense core - diluted halo" density profiles in the AdS bulk were found, with corresponding two-point scalar field correlators obtained within the world line formalism, as a probe of the dual field theory. In this work we cover a much broader free parameter-space of the fermionic solutions in the bulk (from dilute to highly degenerate regime including for the critical point of gravitational collapse), and study their thermodynamic stability by calculating the grand canonical potential and free entropy. Such a stability analysis is performed using the Katz criterion, a technique historically applied to study the gravothermal catastrophe proper of classical self-gravitating systems in flat space. We identify some characteristic features of the unstable regions, both from the bulk and boundary perspectives, that can be used as proxies to detect instabilities on this kind of self-gravitating systems.