There were errors in the way I framed the question last time. So doing a major revision this time.

Consider $SU(2)$ as a homogeneous space $SU(2)\times SU(2)/SU(2)$ and sections of this principle bundle $x \mapsto (g_l(x),g_r(x))$ which are compatible with the projection map $(g_l(x),g_r(x)) \mapsto g_lg_r^{-1} = x$. Hence diagonal action on any section by a map from $SU(2)$ to itself ("gauge function") is compatible with this projection.

Now consider two elements $A$ and $B$ of $SU(2)$ which are acting on the base $SU(2)$ as $x \mapsto AxB^{-1}$. With respect to this a section of the bundle will be called "thermal" (there are physics motivations) if,

$$\sigma(AxB^{-1}) = (A,B).\sigma(x)$$

So the condition of being a thermal section seems to be a guage invariant constraint if one restricts to gauges which have the symmetry that $h(x)=h(AxB^{-1})$.The gauge map acts as $\sigma(x) \mapsto \sigma(x).(h(x),h(x))$.

(All the $.$'s are the standard group multiplication in in $SU(2)\times SU(2)$)

And by the first criteria of what is a valid section all sections can be gauge transformed into one another another since any section giving $(g_l(x),g_r(x))$ is gauge equivalent to the "canonical section" $(I,x^{-1})$ by the gauge function $g_l(x):SU(2)\mapsto SU(2)$ since $g_l(x)g_r(x)^{-1}=x$)

Is there a known mathematically concept equivalent to this? Like any mechanism by which given an $A$ and $B$ and a homogeneous space $G/H$, one would be able to manufacture "thermal" sections for it?

For the specific homogeneous space given it also happens that the push forward of the standard basis in the tangent space at identity of $SU(2)$ ("Pauli matrices") by the thermal section gives a vielbein for the standard metric on $SU(2)$!.

How generic is this?