Quasi‐harmonic lattice‐dynamical calculations are performed to investigate the combined effect of temperature and pressure on the structural and mechanical properties of a prototypical metal–organic framework material: MOF‐5. The softening upon compression of an 𝐴2𝑔 phonon mode at the Γ point in the high‐symmetry F 𝑚3⎯⎯𝑚 structure is identified, which leads to a symmetry reduction and a group–subgroup phase transition to a low‐symmetry F 𝑚3⎯⎯ phase for compressions larger than 0.8%. The effect of the symmetry reduction on the equation‐of‐state of MOF‐5 is investigated, which provides a static bulk modulus K reducing from 17 to 14 GPa and a corresponding change of 𝐾′ (pressure derivative of K) from positive to negative. The effect of pressure on the negative thermal expansion of the framework and on its mechanical response is analyzed. The evolution of the mechanical anisotropy of MOF‐5 as a function of pressure is also determined, which allows identifying the occurrence of a shear‐induced mechanical instability at 0.45 GPa.