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We give a rigorous proof that in any free quantum field theory with a finitegroup global symmetry $\mathrm{G}$, on a compact spatial manifold, atsufficiently high energy, the density of states $\rho_\alpha(E)$ for eachirreducible representation $\alpha$ of $\mathrm{G}$ obeys a universal formulaas conjectured by Harlow and Ooguri. We further prove that this continues tohold in a weakly coupled quantum field theory, given an appropriate scaling ofthe coupling with temperature. This generalizes similar results that werepreviously obtained in $(1+1)$-D to higher spacetime dimension. We discuss therole of averaging in the density of states, and we compare and contrast withthe case of continuous group $\mathrm{G}$, where we prove a universal, albeitdifferent, behavior.

Universal fine grained asymptotics of free and weakly coupled Quantum Field Theory