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.

28 pages, 9 figures. Revised version with additional comments and references, and a section on physical interpretation. Version to appear in CQG