  We have studied the thermodynamic properties of a multiferroic that couples
ferromagnetic and ferroelectric order. Some of the results are independent of
the form of the free energy. We calculate the temperature dependence of the
electric, magnetic, and magnetoelectric susceptibilities. The cross
susceptibility has a temperature dependence related to the mixed (with respect
to E and B) derivatives of the specific heat. The phase transitions are all
second order. In particular, the phase boundary T$_M$(E), where T$_M$ is the
lower magnetic transition as a function of electric field, is described by the
Ehrenfest relation. The magnetoelectric susceptibility is nonzero only below
the lower of the two transition temperatures. We study the properties of the
specific heat, with and without the inclusion of gaussian fluctuations. The
perturbative renormalization group is used to understand the fixed points of
the theory, and we include a discussion of the effect inhomogeneities have for
this model.
