  We consider black holes in an "unsuitable box": a finite cavity coupled to a
thermal reservoir at a temperature different than the black hole's Hawking
temperature. These black holes are described by metrics that are continuous but
not differentiable due to a conical singularity at the horizon. We include them
in the Euclidean path integral sum over configurations, and analyze the effect
this has on black hole thermodynamics in the canonical ensemble. Black holes
with a small deficit (or surplus) angle may have a smaller internal energy or
larger density of states than the nearby smooth black hole, but they always
have a larger free energy. Furthermore, we find that the ground state of the
ensemble never possesses a conical singularity. When the ground state is a
black hole, the contributions to the canonical partition function from
configurations with a conical singularity are comparable to the contributions
from smooth fluctuations of the fields around the black hole background. Our
focus is on highly symmetric black holes that can be treated as solutions of
two-dimensional dilaton gravity models: examples include Schwarzschild,
asymptotically Anti-de Sitter, and stringy black holes.
