Title: Constraining the emergent dark energy models with cosmology-independent observational data

Abstract: In this work, we investigate the phenomenologically emergent dark energy (PEDE) model and its generalized form, namely the generalized emergent dark energy (GEDE) model, which introduces a free parameter \unboldmath {\( \Delta \)} that can discriminate between the \unboldmath{$\mathrm{\Lambda}$}CDM model and the PEDE model. Fitting the emergent dark energy (EDE) models with the observational datasets including the cosmology-independent gamma-ray bursts (GRBs) at high-redshift and the observational Hubble data (OHD), we find a large value of $H_0$ which is close to the results of local measurement of $H_0$ from the SH0ES Collaboration in both EDE models. These results suggest that PEDE and GEDE models can %serve as an important supplement and be possible alternative to the standard cosmological model, pending further theoretical explorations and observational verifications.