The beauty and mysteries of imaginary time and temperature

Yesterday, at UQ Robert Mann gave a nice Quantum Science seminar, "Hot and Cold Accelerating Detectors".

It concerned the Unruh effect: suppose one observer is constantly accelerating relative to another. Then, what is a quantum vacuum (for a free boson or fermion field) to one observer is a thermally populated state to the other.

Specifically, if one considers a field with wave vector k and energy Ω k. Then the expectation of the number operator is,
$$\begin{array}{}\text{(5)}& ⟨<!--\; ⟨\; -->0|b_k^{†<!--\; †\; -->}{b}_k|0⟩<!--\; ⟩\; -->=(e^{2\mathrm{\pi <!--\; \pi \; -->}{\mathrm{\Omega <!--\; \Omega \; -->}}_k/g}-<!--\; -\; -->1{)}^{-<!--\; -\; -->1},\end{array}$$
which corresponds to a Bose distribution with a temperature given by
$$\begin{array}{}& T_{\mathrm{U}\mathrm{n}\mathrm{r}\mathrm{u}\mathrm{h}}=\frac{g}{2\mathrm{\pi <!--\; \pi \; -->}}=\frac{\mathrm{\hslash <!--\; \hslash \; -->}g}{2\mathrm{\pi <!--\; \pi \; -->}c{k}_B}\end{array}$$
where g is the constant acceleration. Note that this formula involves relativity (c), quantum physics (hbar), and statistical mechanics (kB).

I feel there is something rather profound going on here.
Without doing the calculation, it is perhaps not totally surprising that the accelerated observer sees a non-trivial occupation of excited states of the quantum field.
However, what is rather surprising to me is that the state occupation numbers has to be that associated with thermodynamic equilibrium.   Why not some other distribution? And that this holds for both fermions and bosons.
After all, you are starting purely with relativity [and the mathematics of Rindler co-ordinates] and quantum field theory and you are ending up with quantum statistical mechanics.
Is the thermal distribution just a "mathematical accident" because cosh functions are appearing in the Rindler co-ordinates, Bogoliubov transformation of the quantum field, and in thermal distribution?

The Scholarpedia page is helpful, stating this
``can also be understood as a manifestation of the general relationship between temperature and imaginary time in quantum statistical mechanics (KMS theory). When t and τ are extended as complex variables, iτ is revealed as an angular variable in the (it,z) plane. The periodicity determined by g, the acceleration, is the only one that makes functions of τ be analytic in t. That period corresponds, under the KMS theory, to the reciprocal of the Unruh temperature (Dowker, 1978; Christensen and Duff, 1978; Sewell, 1982; Bell et al., 1985; Fulling and Ruijsenaars, 1987).''

I welcome any further elucidations on this rich and subtle issue.