The pressure dependence of the lattice and elastic constants of the orthorhombic YBa ${}_2$ Cu ${}_3$ O ${}_7$ are firstly investigated using the first principles calculations based on the density functional theory. The calculated lattice parameters at 0 GPa are in agreement with the available experimental data. By the elastic stability criteria under isotropic pressure, it is predicted that YBa ${}_2$ Cu ${}_3$ O ${}_7$ with and orthorhombic structure is mechanically stable under pressure up to 100 GPa. On the basis of the elastic constants, Pugh’s modulus ratio, Poisson’s ratio, elastic anisotropy, Debye temperature, and the minimum thermal conductivity of YBa ${}_2$ Cu ${}_3$ O ${}_7$ under pressure up to 100 GPa are further investigated. It is found that its ductility, Debye temperature, and minimum thermal conductivity increase with pressure.