  We describe microcanonical phase transitions and instabilities of the ideal
Fermi gas in general relativity at nonzero temperature confined in the interior
of a spherical shell. The thermodynamic behaviour is governed by the
compactness of rest mass, namely of the total rest mass over radius of the
system. For a fixed value of rest mass compactness, we study the caloric curves
as a function of the size of the spherical box. At low compactness values, low
energies and for sufficiently big systems the system is subject to a
gravothermal catastrophe, which cannot be halted by quantum degeneracy
pressure, and the system collapses towards the formation of a black hole. For
small systems, there appears no instability at low energies. For intermediate
sizes, between two marginal values, gravothermal catastrophe is halted and a
microcanonical phase transition occurs from a gaseous phase to a condensed
phase with a nearly degenerate core. The system is subject to a relativistic
instability at low energy, when the core gets sufficiently condensed above the
Oppenheimer-Volkoff limit. For sufficiently high values of rest mass
compactness the microcanonical phase transitions are suppressed. They are
replaced either by an Antonov type gravothermal catastrophe for sufficiently
big systems or by stable equilibria for small systems. At high energies the
system is subject to the `relativistic gravothermal instability', identified by
Roupas in [1], for all values of compactness and any size.
