  This paper investigates the thermodynamics of a large class of non-Hermitian,
$PT$-symmetric oscillators, whose energy spectrum is entirely real. The
spectrum is estimated by second-order WKB approximation, which turns out to be
very accurate even for small quantum numbers, and used to generate the quantum
partition function. Graphs showing the thermal behavior of the entropy and the
specific heat, at all regimes of temperature, are given. To obtain the
corresponding classical partition function it turns out to be necessary in
general to integrate over a complex "phase space". For the wrong-sign quartic,
whose equivalent Hermitian Hamiltonian is known exactly, it is demonstrated
explicitly how this formulation arises, starting from the Hermitian case.
