  Simulating quantum spin systems at finite temperatures is an open challenge
in many-body physics. This work studies the temperature-dependent spin dynamics
of a pivotal compound, FeI$_2$, to determine if universal quantum effects can
be accounted for by a phenomenological renormalization of the dynamical spin
structure factor $S(\mathbf{q}, \omega)$ measured by inelastic neutron
scattering. Renormalization schemes based on the quantum-to-classical
correspondence principle are commonly applied at low temperatures to the
harmonic oscillators describing normal modes. However, it is not clear how to
extend this renormalization to arbitrarily high temperatures. Here we introduce
a temperature-dependent normalization of the classical moments, whose magnitude
is determined by imposing the quantum sum rule, i.e. $\int d\omega d\mathbf{q}
S(\mathbf{q}, \omega) = N_S S (S+1)$ for $N_S$ dipolar magnetic moments. We
show that this simple renormalization scheme significantly improves the
agreement between the calculated and measured $S(\mathbf{q}, \omega)$ for
FeI$_{2}$ at all temperatures. Due to the coupled dynamics of dipolar and
quadrupolar moments in that material, this renormalization procedure is
extended to classical theories based on SU(3) coherent states, and by
extension, to any SU(N) coherent state representation of local multipolar
moments.
