  When computing the ideal gas thermal canonical partition function for a
scalar outside a black hole horizon, one encounters the divergent
single-particle density of states (DOS) due to the continuous nature of the
normal mode spectrum. Recasting the Lorentzian field equation into an effective
1D scattering problem, we argue that the scattering phases encode non-trivial
information about the DOS and can be extracted by "renormalizing" the DOS with
respect to a reference. This defines a renormalized free energy up to an
arbitrary additive constant. Interestingly, the 1-loop Euclidean path integral,
as computed by the Denef-Hartnoll-Sachdev formula, fixes the reference free
energy to be that on a Rindler space, and the renormalized DOS captures the
quasinormal modes for the scalar. We support these claims with the examples of
scalars on static BTZ, Nariai black holes and the de Sitter static patch. For
black holes in asymptotically flat space, the renormalized DOS is captured by
the phase of the transmission coefficient whose magnitude squared is the
greybody factor. We comment on possible connections with recent works from an
algebraic point of view.
