Three-dimensional hexagonal boron nitride aerogels (hBNAGs) are novel porous materials with many promising applications such as energy storage, thermal insulation and sensing. However, the structure-property relationships of hBNAGs in complicated thermo-mechanical coupled environments are still not clear. In this study, we employed a binary phase-field crystal (PFC) model to construct the atomic structures of hBNAGs, upon which the mechanical and thermal behaviors of hBNAGs were systematically investigated using large-scale atomistic simulations. It is found that the hBNAG geometry and topological defects strongly affect the mechanical and thermal properties. For example, the Young's modulus and tensile strength follow the scaling laws of mass density with a power factor of about 1.4 and 1.2, respectively, indicating that the stretching and bending combine toward tensile deformation. In addition, cracks nucleate around the octagon defects, indicating that the tensile strength is also influenced by the topological defects. Under compression, complicated crumpled deformations and ridges in the entire region are observed and the compression strength follows the scaling law of mass density with a power factor above 2.0, which means that a large portion of the hBNAGs do not contribute to the compression load bearing. We find that hBNAGs have a very low thermal conductivity of about two orders of magnitude lower than that of a hBN sheet. Also, the thermal conductivity of hBNAGs increases with increasing mass density, which also follows a scaling law. The power of the scaling law is about 0.5, indicating that the thermal conductivity has a strong nonlinear dependence on the mass density. Our work provides a deep understanding of the structure-property relationships of hBNAGs, which is useful for the engineering applications of hBNAGs.