It was shown by Tsallis and Cirto that thermodynamical entropy of a gravitational system such as black hole must be generalized to the non-additive entropy, which is given by $S_h=��A^��$, where $A$ is the horizon area and $��$ is the nonextensive parameter \cite{Tsa}. In this paper, by taking the entropy associated with the apparent horizon of the Friedmann-Robertson-Walker (FRW) Universe in the form of Tsallis entropy, and assuming the first law of thermodynamics, $dE=T_hdS_h+WdV$, holds on the apparent horizon, we are able to derive the corresponding Friedmann equations describing the dynamics of the universe with any spatial curvature. We also examine the time evolution of the total entropy and show that the generalized second law of thermodynamics is fulfilled in a region enclosed by the apparent horizon. Then, modifying the emergence proposal of gravity proposed by Padmanabhan and calculating the difference between the surface degrees of freedom and the bulk degrees of freedom in a region of space, we again arrive at the modified Friedmann equation of the FRW Universe with any spatial curvature which is the same as one obtained from the first law of thermodynamics. We also study the cosmological consequences of Tsallis cosmology. Interestingly enough, we find that this model can explain simultaneously the late time acceleration in the universe filled with pressureless matter without invoking dark energy, as well as the early deceleration. Besides, the age problem can be circumvented automatically for an accelerated universe and is estimated larger than $3/2$ age of the universe in standard cosmology. For $��=2/5$, we find $13.12$ Gyr $< t_0 < 16.32 $ Gyr, which is consistent with recent observations. We also comment on the density perturbation in the context of Tsallis cosmology.

11 pages, two columns. A new section regarding the cosmological consequences of this model was added. Also text was revised and new references added