We compute the connected two-time correlator of the maximum $M_N(t)$ of $N$ independent Gaussian stochastic processes (GSP) characterised by a common correlation coefficient $ρ$ that depends on the two times $t_1$ and $t_2$. We show analytically that this correlator, for fixed times $t_1$ and $t_2$, decays for large $N$ as a power law $N^{-γ}$ (with logarithmic corrections) with a decorrelation exponent $γ= (1-ρ)/(1+ ρ)$ that depends only on $ρ$, but otherwise is universal for any GSP. We study several examples of physical processes including the fractional Brownian motion (fBm) with Hurst exponent $H$ and the Ornstein-Uhlenbeck (OU) process. For the fBm, $ρ$ is only a function of $τ= \sqrt{t_1/t_2}$ and we find an interesting ``freezing'' transition at a critical value $τ= τ_c=(3-\sqrt{5})/2$. For $τ< τ_c$, there is an optimal $H^*(τ) > 0$ that maximises the exponent $γ$ and this maximal value freezes to $γ= 1/3$ for $τ>τ_c$. For the OU process, we show that $γ= {\rm tanh}(μ\,|t_1-t_2|/2)$ where $μ$ is the stiffness of the harmonic trap. Numerical simulations confirm our analytical predictions.