  In many rotating fluids, the lowest-order force balance is between gravity,
pressure, and rotational acceleration ('GPR' balance). Terrestrial GPR balance
takes the form of geostrophy and hydrostasy, which together yield the
terrestrial thermal wind equation. By contrast, stellar GPR balance is an
oblateness equation, which determines the departures of the thermal variables
from spherical symmetry; its curl yields the 'stellar thermal wind equation.'
In this sense, the stellar thermal wind should be viewed not as a consequence
of geostrophy, but of baroclinicity in the oblateness. Here we treat the full
stellar oblateness, including the thermal wind, using pressure coordinates. We
derive the generalised stellar thermal wind equation and identify the parameter
regime for which it holds. In the case of the Sun, not considering the full
oblateness has resulted in conflicting calculations of the theoretical
aspherical temperature anomaly. We provide new calculation here and find that
the baroclinic anomaly is ~3-60 times smaller than the barotropic anomaly.
Thus, the anomaly from the thermal wind may not be measurable helioseismically;
but if measurement were possible, this would potentially yield a new way to
bracket the depth of the solar tachocline.
