  The pseudo-fermion representation for $S=1/2$ quantum spins introduces
unphysical states in the Hilbert space which can be projected out using the
Popov-Fedotov trick. However, state-of-the-art implementation of the functional
renormalization group method for pseudo-fermions have so far omitted the
Popov-Fedotov projection. Instead, restrictions to zero temperature were made
and absence of unphysical contributions to the ground-state was assumed. We
question this belief by exact diagonalization of several small-system
counterexamples where unphysical states do contribute to the ground state. We
then introduce Popov-Fedotov projection to pseudo-fermion functional
renormalization, enabling finite temperature computations with only minor
technical modifications to the method. At large and intermediate temperatures,
our results are perturbatively controlled and we confirm their accuracy in
benchmark calculations. At lower temperatures, the accuracy degrades due to
truncation errors in the hierarchy of flow equations. Interestingly, these
problems cannot be alleviated by switching to the parquet approximation. We
introduce the spin projection as a method-intrinsic quality check. We also show
that finite temperature magnetic ordering transitions can be studied via
finite-size scaling.
