  After motivating an interest in the shape of the topological charge density
spectral function in hot Yang-Mills theories, we estimate it with the help of
thermally averaged classical real-time simulations, for $N_{\rm c} = 2,3$.
After subtracting a perturbative contribution at large frequencies, we observe
a non-trivial shape at small frequencies (a dip rather than a peak),
interpolating smoothly towards the sphaleron rate at zero frequency. Possible
frequency scales making an appearance in this shape are discussed. Implications
for warm axion inflation and reheating, and for imaginary-time lattice
measurements of the strong sphaleron rate, are recapitulated.

High Energy Physics - Phenomenology;High Energy Physics - Lattice