  The mineral linarite (PbCuSO$_4$(OH)$_2$) forms a monoclinic structure where
a sequence of Cu(OH)$_2$ units forms a spin-$\frac{1}{2}$ chain. Competing
ferromagnetic nearest-neighbor ($J_1$) and antiferromagnetic
next-nearest-neighbor interactions ($J_2$) in this quasi-one-dimensional spin
structure imply magnetic frustration and lead to magnetic ordering below $T_N
=$2.8 K in a mutliferroic elliptical spin-spiral ground state. Upon the
application of a magnetic field along the spin-chain direction, distinct
magnetically ordered phases can be induced. We studied the thermal conductivity
$\kappa$ in this material across the magnetic phase diagram as well as in the
paramagnetic regime in the temperature ranges 0.07-1 K and 9-300 K. We found
that in linarite the heat is carried mainly by phonons but shows a peculiar
non-monotonic behavior in field. In particular, $\kappa$ is highly suppressed
at the magnetic phase boundaries, indicative of strong scattering of the
phonons off critical magnetic fluctuations. Even at temperatures far above the
magnetically ordered phases, the phononic thermal conductivity is reduced due
to scattering off magnetic fluctuations. The mean free path due to spin-phonon
scattering ($l_{\text{spin-phonon}}$) was determined as function of
temperature. A power law behavior was observed mainly above 0.5 K indicating
the thermal activation of spin fluctuations. In the critical regime close to
the saturation field, $l_{\text{spin-phonon}}$ shows a $1/T$ dependence.
Furthermore, a magnon thermal transport channel was verified in the helical
magnetic phase. We estimate a magnon mean free path which corresponds to about
1000 lattice spacings.
