  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.
