  This paper discusses the Goldreich-Schubert-Fricke instability (GSF) and the
convective overstability (COS) in the context of baroclinic thermal
instabilities in rotating disks around young stars. The vertical shear
instability (VSI) is a global extension of the GSF that affects geometrically
thin disks but follows the same stability criterion. The COS, on the other
hand, also possesses a twin for stellar interiors, specifically, Shibahashi's
vibrational stability of rotating stars. We derive a combined dispersion
relation for GSF and COS with arbitrary cooling times for local perturbations
and determine a new stability criterion beyond the Solberg-H{\o}iland\
criterion. The paper shows that in extension to the stability criterion for the
vertically unstratified case ($N^2_R > 0$), one also needs a barotropic disk
structure to ensure stability towards COS modes. We demonstrate that a
baroclinic disk atmosphere always has a buoyantly unstable direction, although
not necessarily in the radial nor vertical direction. The paper predicts that
for cooling times longer than the critical cooling time for VSI, GSF modes will
always be accompanied by COS modes of similar growth rate. The numerical
companion paper II tests the predictions of growth rates from this paper.
