  We study the phase diagram of the antiferromagnetic $J_1$-$J_2$ Heisenberg
model on the pyrochlore lattice with $S=1$ spins at zero and finite
temperatures. We use a combination of complementary state-of-the-art quantum
many-body approaches such as density matrix renormalization group (DMRG),
density-matrix purification and pseudo-Majorana functional renormalization
group (PMFRG). We present an efficient approach to preserve the applicability
of the PMFRG for spin-1 systems at finite temperatures despite the inevitable
presence of unphysical spin states. The good performance of our methods is
first demonstrated for the nearest-neighbor pyrochlore Heisenberg model where
the finite temperature behavior of the specific heat and uniform susceptibility
show excellent agreement within PMFRG and density-matrix purification.
Including an antiferromagnetic second neighbor coupling we find that the
non-magnetic ground-state phase of the nearest neighbor model extents up to
$J_2/J_1 \sim 0.02 $ within DMRG, beyond which magnetic $\boldsymbol{k}=0$
long-range order sets in. Our PMFRG calculations find the phase transition in a
similar regime $J_2/J_1\sim 0.035(8)$ which, together with the DMRG result,
provides a strong argument for the existence of a small but finite non-magnetic
ground-state phase in the spin-1 pyrochlore Heisenberg model. We also discuss
the origin of discrepancies between different versions of the functional
renormalization group concerning the location of this phase transition.
