We propose a generalised de Sitter scale factor for the cosmology of early and late time universe, including single scalar field is called as inflaton. This form of scale factor has a free parameter q is called as nonextensivity parameter. When q = 1, the scale factor is de Sitter. This scale factor is an intermediate form between power-law and de Sitter. We study cosmology of such families. We show that both kinds of dark components, dark energy and dark matter simultaneously are described by this family of solutions. As a motivated idea, we investigate inflation in the framework of q-de Sitter.We consider three types of scenarios for inflation. In a single inflation scenario, we observe that, inflation ended without any specific ending inflation φend , the spectral index and the associated running of the spectral index are ns − 1 ∼ −2, αs ≡ 0. To end the inflation: we should have q = 34 . We deduce that the inflation ends when the evolution of the scale factor is a(t) = e3/4(t). With this scale factor there is no need to specify φend . As an alternative to have inflation with ending point, We will study q-inflation model in the context of warm inflation. We propose two forms of damping term . In the first case when = 0, we show the scale invariant spectrum, (Harrison-Zeldovich spectrum, i.e. ns = 1) may be approximately presented by(q = 9 10, N = 70). Also there is a range of values of R and ns which is compatible with the BICEP2 data where q = 9 10 . In case = 1V (φ), it is observed that small values of a number of e-folds are assured for small values of q parameter. Also in this case, the scaleinvariant spectrum may be represented by (q,N) = ( 9 10 , 70). For q = 9 10 a range of values of R and ns is compatible with the BICEP2 data. Consequently, the proposal of q-de Sitter is consistent with observational data.We observe that the non-extensivity parameter q plays a significant role in inflationary scenario.
