Summary: In this study, we examine on the expansion history of the universe from the inflationary era to the late-time phantom era in the framework of higher dimensional \(F(R)\) gravity. Following the unified first law of thermodynamics method, we find a general entropy expression of the gravity theory for the apparent horizon of the universe. Based on the assumption that the dark energy mystery is closely related to the dimensions of spacetime, we research the progress eras of the universe according to the opening of the space-time dimensionality. In a more general framework, we consider the space time manifold as \(n + 1\)-dimensional spherically symmetric form and discuss the phases of the universe through the validity of entropy. During the inflation, we obtain a 10 dimensional spacetime model which is estimated by the string theory. However, while in the radiation era it is obtained 5 dimensional Kaluza-Klein spacetime, in the dust era 4 spacetime dimensions within the framework of entropy keep standing. At small curvatures of spacetime, a process occurs in which a metric deformation is observed with the negative value of the equation of the state (EoS) parameter, \(w < -\frac{1}{3}\). This leads to the release of dark energy from the extra dimensions resulting in a 10 dimensional universe. Since the currently observable universe is in four dimensional spacetime form, this fact indicates that the universe needs to pass into a new phase. Hence, de Sitter phase \(w = -1\) of the universe gives 10 dimensional spherically symmetrical spacetime, but here the deformation disappears. Therewithal, we find that 4 dimensional universe model emerges in the phantom region without disturbing the spherical and symmetrical structure of the universe. This means that the late-time accelerated expansion of the universe could be due to dark energy emanating from the extra dimensions.