We demonstrate that there appear finite-time future singularities in f(T) gravity with T being the torsion scalar. We reconstruct a model of f(T) gravity with realizing the finite-time future singularities. In addition, it is explicitly shown that a power-law-type correction term T$\beta${} ($\beta${}>1) such as a T2 term can remove the finite-time future singularities in f(T) gravity. Moreover, we study f(T) models with realizing inflation in the early universe, the $\Lambda${}CDM model, little rip cosmology and pseudo-rip cosmology. It is demonstrated that the disintegration of bound structures for little rip and pseudo-rip cosmologies occurs in the same way as in gravity with corresponding dark energy fluid. We also discuss that the time-dependent matter instability in the star collapse can occur in f(T) gravity. Furthermore, we explore thermodynamics in f(T) gravity and illustrate that the second law of thermodynamics can be satisfied around the finite-time future singularities for the universe with the temperature inside the horizon being the same as that of the apparent horizon.

We reconstruct F (R) gravity models with exponential and power-law forms of the scale factor in which bounce cosmology can be realized. We explore the stability of the reconstructed models with analyzing the perturbations from the background solutions. Furthermore, we study an F (R) gravity model with a sum of exponentials form of the scale factor, where the bounce in the early universe as well as the late-time cosmic acceleration can be realized in a unified manner. As a result, we build a second order polynomial type model in terms of R and show that it could be stable. Moreover, when the scale factor is expressed by an exponential form, we derive F (R) gravity models of a polynomial type in case of the non-zero spatial curvature and that of a generic type in that of the zero spatial curvature. In addition, for an exponential form of the scale factor, an F (R) bigravity model realizing the bouncing behavior is reconstructed. It is found that in both the physical and reference metrics the bouncing phenomenon can occur, although in general the contraction and expansion rates are different each other.

We investigated the regular multihorizon black holes in the Einstein gravity, $F(R)$ gravity, and the 5 dimensional Gauss-Bonnet gravity, all of them coupled with nonlinear electrodynamics. We presented several explicit examples of the actions which admit the solutions describing regular black hole space-time with multihorizons. Thermodynamics of the obtained black hole solutions is studied. The explicit expressions of the temperature, the entropy, the thermodynamical energy and the free energy are obtained. Although the temperature vanishes in the extremal limit where the radii of the two horizons coincide with each other as in the standard multihorizon black hole like the Reissner-Nordström black hole or the Kerr black hole, the larger temperature corresponds to the larger horizon radius. This is different from the standard black holes, where the larger temperature corresponds to the smaller horizon radius. We also found that the specific heat becomes positive for the large temperature, which is also different from the standard black holes, where the specific heat is negative. It should be also noted that the thermodynamical energy is not identical with the ADM mass. Furthermore in case of the Gauss-Bonnet gravity,it is demonstrated that the entropy can become negative.