There seem to be misunderstandings inherent in the original question.
Surely the *other* thermodynamic potentials (and the entropy) all
change. The Helmholtz free energy, in particular, is the one which
should be applied in this case. The internal energy does not change,
as Joel correctly points out.
Given all the perfectly well understood thermodynamics that have
been produced in the last 200 years, do we really need a new
function, "thermal energy"? Is there any conceivable value to it?
The function that most nearly approaches "thermal energy" in my
view is the enthalpy*, but do we need another name for the enthalpy?
"Thermal energy" represents a problem, not a solution. Let's get
rid of it entirely!
Leigh
*The enthalpy change in any process carried out at constant pressure
(say the ambient pressure) is equal to the heat transferred to an
unconstrained system in a reversible process (that is, the system
remains in thermal equilibrium during the heat transfer). It is not
equal to the change in internal energy if the volume of the system
changes in the process. Here I have considered hydrostatic systems
only.