  We report a study on the thermal conductivity of CuFe$_{1-x}$Ga$_x$O$_2$ ($x
=$ 0--0.12) single crystals at temperatures down to 0.3 K and in magnetic
fields up to 14 T. CuFeO$_2$ is a well-known geometrically frustrated
triangular lattice antiferromagnet and can be made to display multiferroicity
either by applying magnetic field along the $c$ axis or by doping nonmagnetic
impurities, accompanied with rich behaviors of magnetic phase transitions. The
main experimental findings of this work are: (i) the thermal conductivities
($\kappa_a$ and $\kappa_c$) show drastic anomalies at temperature- or
field-induced magnetic transitions; (ii) the low-$T$ $\kappa(H)$ isotherms
exhibit irreversibility in a broad region of magnetic fields; (iii) there are
phonon scattering effect caused by magnetic fluctuations at very low
temperatures. These results demonstrate strong spin-phonon coupling in this
material and reveal the non-negligible magnetic fluctuations in the "ground
state" of pure and Ga-doped samples.
