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Structural phase transition, spontaneous polarization, piezoelectric response, and band structure of aluminum nitride under hydrostatic pressure are systematically studied via first-principles calculations. The band structures are obtained from the HSE06 range-separated hybrid functional. Our calculated results exhibit interesting behaviors: (i) Just like the cases of uniaxial and in-plane strains, the material undergoes a structural transition from the equilibrium wurtzite phase to a pseudographitic h-MgO phase at large pressure. (ii) Although the new phase is nonpolar, the spontaneous polarization of wurtzite phase is greatly enhanced by pressure and reaches the maximum value at the phase transition. (iii) The appropriately applied pressure remarkably enhances the piezoelectric response for wurtzite phase, with the strongest behavior appearing at the phase transition. This is consistent, in that the wurtzite structure becomes markedly soft along the polar axis as pressure increases and similar to the structural transition of perovskites from ferroelectric to paraelectric phases. (iv) The wurtzite phase under pressure undergoes a direct-indirect bandgap transition, with the conduction band minimum (CBM) no longer at zone center Γ, but at the zone-edge K point and the valence band maximum (VBM) at Γ. In addition, the polar-nonpolar structural transformation simultaneously gives rise to another bandgap transition from indirect to indirect with the CBM shifting from K to M point, but the VBM still at Γ. This is remarkably different from the results of uniaxial and in-plane strains. The anomalous behaviors of the band structure originate from that the top valence and bottom conduction states at different reciprocal lattice points show the different dependence on hydrostatic pressure.