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Cobalt oxyhydroxide (HCoO2), which belongs to the heterogenite mineral family, exists in two hexagonal polytypes (2H) and rhombohedral (3R). Here, we investigate the high pressure behaviour of both polytypes of HCoO2 using state‐of‐the‐art first principles calculations‐based density functional theory. We have used both local density approximation (LDA) and generalised gradient approximation (GGA) for exchange correlation functional. For the calculation of electronic band structures, we have also used the on‐site Coulomb interaction U term with LDA calculations and found improvement in results. The obtained ground state properties for both polytypes agree well with experimental and previous theoretical data. The electronic band structure shows that both phases are semiconductor with an indirect band gap in the ran\ge of 2.01–2.06 eV. The pressure‐dependent phonon dispersion curves and elastic constants depict the instability of 2H and 3R phases around 35 and 40 GPa, respectively. The O‐H distance gradually decreases for both phases with pressure and results into the less compressibility in z‐direction. The theoretically calculated pressure‐dependent Raman spectra of 2H‐HCoO2 show the peak broadening nature of E2g mode with pressure whereas in the case of 3R‐HCoO2, the A1g mode disappears beyond 45 GPa pressure. This indicates the existence of phase transition beginning around this pressure, which is driven by the shear modulus. A detailed analysis of the mode compatibility in the case of 2H polytypes is also reported. The phonon mode evolution with pressure is also analysed.