  The thermodynamical entropy of a system which consists of different kinds of
ideal gases is known to be defined successfully in the case when the
differences are described by classical or quantum theory. Since these theories
are special examples in the framework of generalized probabilistic theories
(GPTs), it is natural to generalize the notion of thermodynamical entropy to
systems where the internal degrees of particles are described by other possible
theories. In this paper, we consider thermodynamical entropy of mixing in a
specific series of theories of GPTs called the regular polygon theories, which
can be regarded from a geometrical perspective as intermediate theories between
a classical trit and a quantum bit with real coefficients. We prove that the
operationally natural thermodynamical entropy of mixing does not exist in those
inbetween theories, that is, the existence of the natural entropy results in
classical and quantum-like theories among the regular polygon theories.
