We study early universe with a particular form of F(T) teleparallel gravity theory, in which inflation is driven by a scalar field. To ensure slow rollover, two different potentials are chosen in a manner, such that they remain almost flat for large initial value of the scalar field. Inflationary parameters show wonderful fit with the presently available Planck's data set. The energy scale of inflation is sub-Planckian, and graceful exit from inflation is also administered. The chosen form of F(T) administers late-time cosmic acceleration too. In the process, unification of the early inflation with late-time acceleration is ensured. Unfortunately, a decelerated radiation-dominated era is only possible with a different form of (quartic) potential, which being devoid of a flat section does not admit slow rollover.

As in the case of Lanczos–Lovelock gravity, the main advantage of [Formula: see text] gravity is said to be that it leads to second-order field equations, while [Formula: see text] gravity theory leads to fourth-order equations. We show that it is rather a disadvantage, since it leads to the unresolved issue of ‘Branched Hamiltonian’. The problem is bypassed in [Formula: see text] gravity theory.

Under conformal transformation, [Formula: see text] theory of gravity in Palatini formalism leads to a Brans–Dicke type of scalar-tensor equivalent theory with a wrong sign in the effective kinetic energy term. This means that the effective scalar acts as the dark energy and so late-time cosmic acceleration in the matter-dominated era is accountable. However, we unveil some aspects of Palatini formalism, which reveals the fact that the formalism is not suitable to explain the cosmological evolution of the early universe with [Formula: see text] gravity alone. Additionally, it is noticed that some authors, in an attempt to explore Noether symmetry of the theory changed the sign of the kinetic term and hence obtained the wrong answer. Here, we make the correction and unmask a very interesting aspect of symmetry analysis. Mathematical inequivalence between Jordan’s and Einstein’s frame in Palatini [Formula: see text] theory has also been revealed.

[Formula: see text] theory of gravity is claimed to admit a host of conserved currents under the imposition of Noether symmetry following various techniques. However, for a constrained system such as gravity, Noether symmetry is not on-shell. As a result, the symmetries do not necessarily satisfy the field equations in general, constraints in particular, unless the generator is modified to incorporate the constraints. In this paper, we apply the first theorem of Poisson to unveil the fact that not all the conserved currents appearing in the literature for [Formula: see text] theory of gravity satisfy the field equations. We also provide a list of available forms of [Formula: see text] along with associated conserved currents and construct a generalized action, which might address the cosmic puzzle, elegantly.