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In this work, we study the extended viscous dark energy models in the contextof matter perturbations. To do this, we assume an alternative interpretation ofthe flat Friedmann-Lema\^itre-Robertson-Walker Universe, through thenonadditive entropy and the viscous dark energy. We implement the relativisticequations to obtain the growth of matter fluctuations for a smooth version ofdark energy. As result, we show that the matter density contrast evolvessimilarly to the $\Lambda$CDM model in high redshift; in late time, it isslightly different from the standard model. Using the latest geometrical andgrowth rate observational data, we carry out a Bayesian analysis to constrainparameters and compare models. We see that our viscous models are compatiblewith cosmological probes, and the $\Lambda$CDM recovered with a $1\sigma$confidence level. The viscous dark energy models relieve the tension of $H_0$in $2 \sim 3 \sigma$. Yet, by involving the $\sigma_8$ tension, some models canalleviate it. In the model selection framework, the data discards the extendedviscous dark energy models.

Growth of matter perturbations in the extended viscous dark energy models