  We use our recently developed functional renormalization group (FRG) approach
for quantum spin systems to investigate the phase diagram of the frustrated
$J_{1}J_{2}J_{3}$ quantum Heisenberg model on a cubic lattice. From a simple
truncation of the hierarchy of FRG flow equations for the irreducible
spin-vertices which retains only static spin fluctuations and neglects the flow
of the four-spin interaction, we can estimate the critical temperature with a
similar accuracy as the numerically more expensive pseudofermion FRG. In the
regime where the ground state exhibits either ferromagnetic or
antiferromagnetic order, a more sophisticated truncation including the
renormalization of the four-spin interaction as well as dynamic spin
fluctuations reveals the underlying renormalization group fixed point and
yields critical temperatures which deviate from the accepted values by at most
4 %.
