Gamma-ray bursts (GRBs) are among the most powerful transient phenomena in the Universe. Their complex light curves may encode signatures of their emission mechanisms, which remain largely uncovered. This work presents the first extended redshift-corrected analysis of the average power-density spectrum (PDS) of 214 GRBs observed by the Gamma-ray Burst Monitor on the Fermi Gamma-Ray Space Telescope between 2008 and 2023. Our results reveal a strong power-law behaviour in the PDS, with high-frequency power-law slopes clustering around $-1.9 \pm 0.2$ for long GRBs, largely independent of GRB brightness, duration, and redshift. The analysis of the short GRBs remains limited by redshift data, but it is shown that they exhibit comparable slopes to the long bursts when not redshift-corrected. For the first time, we were also able to study the average PDS of the different GRB phases. Our findings show that the precursor emission may differ from the prompt emission, suggesting different physical origins. Comparisons with noise profiles confirm that the observed power-law features are intrinsic to the GRB signal in most cases.