  The Bekenstein-Hawking entropy suggests that thermodynamics is an intrinsic
ingredient of gravity. Here, we explore the idea that requirements of
thermodynamic consistency could determine the gravitational entropy in other
set-ups. We implement this idea in a simple model: static, spherically
symmetric solutions to Einstein's equations corresponding to self-gravitating
radiation. We find that the principle of maximum entropy provides a consistent
thermodynamic description of the system, only if the entropy includes a
contribution from the spacetime singularities that appear in the solutions of
Einstein's equations. The form of the singularity entropy is stringently
constrained from consistency requirements, so that the existence of a simple
expression satisfying these constraints is highly non-trivial, and suggests of
a fundamental origin. We find that the system is characterized by three
equilibrium phases, and we conduct a preliminary investigation of the
associated phase transitions. These results demonstrate the point that
gravitational entities other than horizons are endowed with thermodynamic
properties.
