The elastic constants of δ-AlOOH have been calculated for applied pressures in the range 0-40 GPa using three exchange-correlation functionals, i.e. PBEsol, TCAsol and PBE. PBEsol and TCAsol, which describe the geometry of this system very well, give similar results, while the PBE values are smaller, a consequence of the fact that the PBE lattice parameters are too large. In agreement with the experiments, the calculations show that at low pressure the axial compressibility is greater in the b direction and smaller in the c direction, while the opposite is true at high pressure. This change takes place 5 GPa before the calculated hydrogen bond symmetrization transition. Combining this result with those reported in the experimental papers, one obtains a strong indication that symmetrization takes place at about 15 GPa, in excellent agreement with the PBEsol and TCAsol predictions.