Modified cosmology from extended entropy with varying exponent

We present a modified cosmological scenario that arises from the application of non-extensive thermodynamics with varying exponent. We extract the modified Friedmann equations, which contain new terms quantified by the non-extensive exponent, possessing standard $\Lambda$CDM cosmology as a subcase. Concerning the universe evolution at late times we obtain an effective dark energy sector, and we show that we can acquire the usual thermal history, with the successive sequence of matter and dark-energy epochs, with the effective dark-energy equation-of-state parameter being in the quintessence or in the phantom regime. The interesting feature of the scenario is that the above behaviors can be obtained even if the explicit cosmological constant is set to zero, namely they arise purely from the extra terms. Additionally, we confront the model with Supernovae type Ia and Hubble parameter observational data, and we show that the agreement is very good. Concerning the early-time universe we obtain inflationary de Sitter solutions, which are driven by an effective cosmological constant that includes the new terms of non-extensive thermodynamics. This effective screening can provide a description of both inflation and late-time acceleration with the same parameter choices, which is a significant advantage.