Yes. It turns out that your $T_L$ is equal to $-T/\omega$, where $\omega$ is the angular velocity and $T$ is the usual temperature. We normally work with the reciprocals of such quantities, and in the language of non-equilibrium thermodynamics we say that a gradient in $-\omega/T$ is the "thermodynamic force conjugate to" a flow of angular momentum.
Within the formalism of thermodynamics itself there is indeed nothing special about energy. (There is, however, quite a lot that's special about energy when it comes to mechanics.) However, the usual terminology and notation obscures this quite a bit.
We usually write the fundamental equation of thermodynamics with the energy on the left-hand-side, like this:
$$
dU = TdS - pdV + \sum_i \mu_i dN_i.
$$
This equation can be extended with many other terms, including $\phi dQ$ (electric potential times change in charge) and $\omega dL$ (angular velocity times change in angular momentum). However, the "special" quantity here is the entropy, $S$, which is non-decreasing while all the other extensive quantities are conserved. We can rearrange this to put the special quantity on the left, and to get
$$
dS = \frac{1}{T} dU + \frac{p}{T}dV - \sum_i \frac{\mu_i}{T}dN_i + \dots - \frac{\phi}{T}dQ - \frac{\omega}{T} dL.
$$
This observation is the basis of non-equilibrium thermodynamics. It follows immediately from this that
$$
\frac{\partial S}{\partial L} = -\frac{\omega}{T}.
$$
It also follows that angular momentum cannot be spontaneously transferred from one body to another while keeping all other quantities constant unless the second body has greater $-{\omega}/{T}$.
However, that "while keeping other quantities constant" is a bit tricky. In just about any reasonable situation, adding angular momentum to a system will also change its energy. The same is true of changes in volume, chemical composition or charge: changing these things will, generally speaking, also change the energy. This is probably the main historical reason why energy is seen as special in thermodynamics: it's the only thing you can practially change while keeping everything else constant. (We call this "heating up" or "cooling down" a system.)
So while it is quite possible to define angular-momentum analogues of heat, free energy and the Carnot limit, these don't tend to have the same immediate practical applications as the energy-based versions. Nevertheless, I think the existence of such quantities is an enlightening and often-overlooked observation. I would encourage you to keep on thinking along these lines, since understanding the symmetry between energy and the other conserved quantities leads to a deeper understanding of thermodynamics as a whole.
