  This work investigates how a conical singularity can affect the specific heat
of systems. A free nonrelativistic particle confined to the lateral surface of
a cone -- conical box -- is taken as a toy model. Its specific heat is
determined as a function of the deficit angle and the temperature. For a
vanishing deficit angle, the specific heat is that of a particle in a flat disk
where a characteristic temperature separates quantum and classical behaviors,
as usual. By increasing the deficit angle the characteristic temperature
increases also, and eventually another characteristic temperature (which does
not depend on the deficit angle) arises. When the cone gets sufficiently sharp,
at low and intermediate temperatures the azimuthal degree of freedom is
suppressed. At low temperatures the specific heat varies discontinuously with
the deficit angle. Connections between certain theorems regarding common zeros
of the Bessel functions and this discontinuity are reported.
