Aluminum oxide hydroxide, AlOOH, is a prototypical hydrous mineral in the geonomy. The study of the high-pressure phase evolution of AlOOH is of fundamental importance in helping to understand the role of hydrous minerals in the water storage and transport in Earth, as in other planets. Here, we have systematically investigated the high-pressure phase diagram of AlOOH up to 550 GPa using the efficient crystal structure analysis by particle swarm optimization (CALYPSO) algorithm in conjunction with first principles calculations. We predict a peculiar monoclinic phase (space group P21/c, 16 atoms/cell, Z=4 ) as the most stable phase for AlOOH above 340 GPa. The occurrence of this new phase results in the breakup of symmetric linear O-H-O hydrogen bonds into asymmetric, bent O-H-O linkages and in sevenfold coordinated metal cations. The new P21/c phase turns out to be a universal high-pressure phase in group 13 oxide hydroxides, and stable for both compressed GaOOH and InOOH. The formation of the new phase in all compounds is favored by volume reduction due to denser packing.