  The magnon propagation length ($\langle\xi\rangle$) of a ferro/ferrimagnet
(FM) is one of the key factors that controls the generation and propagation of
thermally-driven spin current in FM/heavy metal (HM) bilayer based
spincaloritronic devices. Theory predicts that for the FM layer,
$\langle\xi\rangle$ is inversely proportional to the Gilbert damping ($\alpha$)
and the square root of the effective magnetic anisotropy constant ($K_{\rm
eff}$). However, direct experimental evidence of this relationship is lacking.
To experimentally confirm this prediction, we employ a combination of
longitudinal spin Seebeck effect (LSSE), transverse susceptibility, and
ferromagnetic resonance experiments to investigate the temperature evolution of
$\langle\xi\rangle$ and establish its correlation with the effective magnetic
anisotropy field, $H_K^{\rm eff}$ ($\propto K_{\rm eff}$) and $\alpha$ in
Tm$_3$Fe$_5$O$_{12}$ (TmIG)/Pt bilayers. We observe concurrent drops in the
LSSE voltage and $\langle\xi\rangle$ below 200$^\circ$K in TmIG/Pt bilayers
regardless of TmIG film thickness and substrate choice and attribute it to the
noticeable increases in $H_K^{\rm eff}$ and $\alpha$ that occur within the same
temperature range. From the TmIG thickness dependence of the LSSE voltage, we
determined the temperature dependence of $\langle\xi\rangle$ and highlighted
its correlation with the temperature-dependent $H_K^{\rm eff}$ and $\alpha$ in
TmIG/Pt bilayers, which will be beneficial for the development of rare-earth
iron garnet-based efficient spincaloritronic nanodevices.
