  We analyze the thermodynamics of spherically symmetric thin-shell solutions
to Einstein's equations, including solutions with negative interior mass. We
show the inclusion of such solutions is essential for the thermodynamic
consistency of the system: the Maximum Energy Principle applies when we include
an entropy term from the singularity of the negative-mass solutions, in
addition to the Bekenstein-Hawking term for the entropy of solutions with
positive interior mass. Then, the thermodynamic analysis leads to four distinct
thermodynamic phases. We also show that all types of solutions can be either
thermodynamically stable or dynamically stable, but only solutions with zero
interior mass can be both. Since most of our results are analytic, thin shell
models emerge as a useful theoretical paradigm for exploring gravitational
thermodynamics. Our results provide an additional argument in support of the
assignment of entropy to the singularity of negative-mass Schwarzschild
spacetimes, and, consequently, to Penrose's conjecture about the assignment of
entropy to singularities.
