How can Rovelli's thermal time hypothesis be consistent with the apparent existence of a privileged time in the sense of cosmology?
As you know, within the thermal time approach you start from a probability distribution function (or density matrix in the quantum setting) $\rho$, you derive an Hamiltonian (or Hamiltonian operator) $H=-T\,\ln\rho$ and define the time $t$ as the variable with respect to which the system evolves given the Hamiltonian $H$.
Now, $\rho$ can be really any physically meaningful distribution density, and different $\rho$'s will give rise to different thermal times. In other words, $t$ is defined with respect to the observer that measures the distribution $\rho$. If as your $\rho$ you choose the distribution which describes a cosmological fluid in the cosmological frame of reference, then you should obtain as your thermal time the "unique" cosmological time referred to in your question. In Rovelli's book "Quantum Gravity" (sec. 3.4) it is stated that
The thermal time hypothesis works surprisingly well in a number of cases. For example, if we start from a radiation-filled [...] cosmological model [...] and write a statistical state representing the cosmological background radiation, then the thermal time of this state turns out to be precisely the Friedmann time.
This "Friedmann" or "cosmological" time is privileged in the following sense: it is just the appropriate time variable to use when describing the evolution of the cosmological fluid. In other words, again, you are starting from a (matter/radiation/etc.) distribution to find the appropriate time variable, which is exactly what you do (albeit following a very, very different path) in the thermal time approach. Therefore the thermal time hypothesis is completely consistent with the existence of a (not so) privileged cosmological time.
I cannot disagree about my need to make the distinction you noted above, but that is not inappropriate on my part, as you so claim, due to this in Wikipedia:
Even though the words 'hypothesis' and 'theory' are often used synonymously, a scientific hypothesis is not the same as a scientific theory...." 4Sept2018.
The so-called "arrow of time" has always pointed toward the future, yet the future does not exist, but even if it did, it is absurd to say time flows toward the future. How could anyone make such a silly claim? Time passes, alright, but only in the present-time because the past does exist in our physical reality either. Plus, time is a property only of discrete matter, like gravity is a property of all masses. Like gravity, time is a fundamental force, too. It has the power to age all discrete matter inversely proportional to its speed, which explains the time dilation phenomenon. Fundamental forces do not interact with each other, per se, yet gravity can indirectly affect time rates due to the gravity potential increasing the speeds of matter in free fall, which acts to decrease their time rates.
The thermal time hypothesis is not yet a theory since it makes no predictions nor does it present any falsifiable premises. Thus, I believe there is only cosmological time - if we prefer to call it that.