Resistive and ballistic phonon transport in <i>β</i> - <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:mrow><mml:msub><mml:mi mathvariant="normal">Ga</mml:mi><mml:mn>2</mml:mn></mml:msub><mml:msub><mml:mi mathvariant="normal">O</mml:mi><mml:mn>3</mml:mn></mml:msub></mml:mrow></mml:mat

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.