  We numerically study the evolution of spectral correlations in the
entanglement Hamiltonian (EH) of non-interacting fermions in the
Aubry-Andr\'e-Harper (AAH) model. We analyze the time evolution of the EH
spectrum in a nonequilibrium setting by studying several quantities: spectral
distribution, level statistics, entanglement entropy, and spectral form factor
(SFF) in the context of the delocalization-localization transition in the AAH
model. It is observed that the SFF of the entanglement spectrum in the
delocalized phase and at the phase-transition point evolves in three-time
intervals. We make a systematic study of the emergence of these three
timescales for various initial states and find that the number of time
intervals remains three unless the Hamiltonian is tuned in the localized phase
or when the initial state is maximally entangled, then there is a featureless
time evolution. We find a broad direct correlation between the entanglement
entropy and the length of the ramp of the SFF. We also find that in the
delocalized phase the spectral correlations are stronger in the center of the
spectrum and grow progressively weaker as more and more of the spectrum is
considered.
