  The anisotropic thermal conductivity and the phonon mean free path (mfp) in
monoclinic $\beta$-Ga$_2$O$_3$ single crystals and homoepitaxial films of
several $\mu$m were determined using the 3$\omega$-method in the temperature
range from 10K-300 K. The measured effective thermal conductivity of both,
single crystal and homoepitaxial films are in the order of 20 W/(mK) at room
temperature, below 30 K it increases with a maximum of 1000 to 2000 W/(mK) and
decreases with T$^3$ below 25 K. Analysis of the phonon mfp shows a dominance
of phonon-phonon-Umklapp scattering above 80 K, below which the influence of
point-defect scattering is observed. Below 30 K the phonon mfp increases until
it is limited by the total $\beta$-Ga$_2$O$_3$ sample size. A crossover from
resistive to ballistic phonon transport is observed below 20 K and boundary
effects of the total sample size become dominant. This reveals that the
homoepitaxial film-substrate interface is highly phonon-transparent. The
resistive and ballistic phonon transport regimes in $\beta$-Ga$_2$O$_3$ are
discussed corresponding to the models of Callaway and Majumdar, respectively.
