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Boron-based 2D materials are of current interest. However, graphene-like geometry is unstable for B due to the electron deficiency, which can be stabilized by introducing H, F and Cl. Here, using density functional theory combined with phonon Boltzmann transport equation, we perform systematic studies on how the functionalization changes the lattice thermal conductivity (LTC). We find that when going from hydrogenation to fluorination and chlorination, the LTC along zigzag direction changes from 367.6 to 211.3 and 43.0 W/(m·K), while the corresponding values in armchair direction are 279.6, 198.9, and 41.6 W/(m·K), respectively. These huge differences imply the sensitivity of LTC to functionalization, which can be attributed to the enhanced anharmonicity as revealed by analyzing group velocity, Gruneisen parameter, anharmonic scattering rates, and three-phonon scattering space.