  We study the scalar-tensor theory of gravity profoundly in the action level
as well as in the thermodynamic level. Contrary to the usual description in the
literature about the equivalence in the two conformally connected frames, this
paper addresses several incomplete inferences regarding it and mentions some
inequivalences which were not pointed out earlier. In the thermodynamic level,
our analysis shows the two frames are equivalent. In that process, we identify
the entropy, the energy and the temperature for the thermodynamic description,
and we find these quantities are conformally invariant even without any prior
assumption. The same conclusion is reached from the gravitational action as
well as from the Gibbons-Hawking-York boundary term, establishing the result in
a more convincing manner.
