Using a first principles method, we have investigated the crystal structure of a high-pressure phase of AlOOH (δ-AlOOH). It is an important substance in subducting sedimentary rocks, and could possibly serve as a water reservoir in the Earth's lower mantle. Three types of stable structure were obtained with various hydrogen positions. Two of them have an asymmetric hydrogen bond and the other a symmetric one. The asymmetric hydrogen bond of the low-pressure phase transforms to the symmetrized one under high pressure. The symmetrized phase has a much larger bulk modulus than the asymmetrized phases. The optimized structures and their respective bulk moduli obtained from our calculation indicate that the subtle change of hydrogen position causes a significant effect on the hydrogen bonding nature and elastic properties of δ-AlOOH.