We propose an approach considering the nonextensive effects in the context of the Verlinde theory in order to address an extended cosmological model in the context of viscous dark energy. Specifically, this model leads to a tiny perturbation in the dynamics of the expansion of the universe through the generalized Friedmann equations so-called the extended ΛCDM model. From the observational test standpoint, we make a Bayesian analysis of the models of bulk viscosity for dark energy which follows the Eckart theory of bulk viscosity. These models are investigated through the context of both models ΛCDM and extended ΛCDM. The Bayesian analysis is performed using the data of CMB Distance priors, Baryon Acoustic Oscillations Measurements, Cosmic Chronometers, and SNe Ia distance measurements.

In this work, we study the extended viscous dark energy models in the context of matter perturbations. To do this, we assume an alternative interpretation of the flat Friedmann–Lemaître–Robertson–Walker Universe, through the nonadditive entropy and the viscous dark energy. We implement the relativistic equations to obtain the growth of matter fluctuations for a smooth version of dark energy. As result, we show that the matter density contrast evolves similarly to the $$\Lambda $$ Λ CDM model in high redshift; however, in late time, it is slightly different from the standard model. Using the latest geometrical and growth rate observational data, we carry out a Bayesian analysis to constrain parameters and compare models. We see that our viscous models are compatible with cosmological probes, and the $$\Lambda $$ Λ CDM recovered with a $$1\sigma $$ 1 σ confidence level. The viscous dark energy models relieve the tension of $$H_0$$ H 0 in $$2 \sim 3 \sigma $$ 2 ∼ 3 σ . Yet, by involving the $$\sigma _8$$ σ 8 tension, some models can alleviate it. In the model selection framework, the data discards the extended viscous dark energy models.

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