We studied bulk crystals of wurtzite AlN by means of uniaxial pressure-dependent Raman measurements. As a result, we derive the phonon pressure coefficients and deformation potentials for all zone center optical phonon modes. For the A1 and E1 modes we further experimentally determined the uniaxial pressure dependence of their longitudinal optical -transverse optical (LO-TO) splittings. Our experimental approach delivers new insight into the large variance among previously reported phonon deformation potentials, which are predominantly based on heteroepitaxial growth of AlN and the ball-on-ring technique. Additionally, the measured phonon pressure coefficients are compared to their theoretical counterparts obtained by density functional theory implemented in the SIESTA package. Generally, we observe a good agreement between the calculated and measured phonon pressure coefficients but some particular Raman modes exhibit significant discrepancies similar to the case of wurtzite GaN and ZnO, clearly motivating the presented uniaxial pressure-dependent Raman measurements on bulk AlN crystals.