Along this review, we focus on the study of several properties of modified gravity theories, in particular on black-hole solutions and its comparison with those solutions in General Relativity, and on Friedmann-Lemaître-Robertson-Walker metrics. The thermodynamical properties of fourth order gravity theories are also a subject of this investigation with special attention on local and global stability of paradigmatic f (R) models. In addition, we revise some attempts to extend the Cardy-Verlinde formula, including modified gravity, where a relation between entropy bounds is obtained. Moreover, a deep study on cosmological singularities, which appear as a real possibility for some kind of modified gravity theories, is performed, and the validity of the entropy bounds is studied.

We discuss conformal issues of pure and extended teleparallel gravity. In particular, we present formulations of conformal transformation in teleparallel gravity. Furthermore, we propose conformal scalar and gauge field theories in teleparallel gravity and study conformal torsion gravity. We explicitly demonstrate that a power-law acceleration (including the $\Lambda$CDM universe) as well as the de Sitter expansion of the universe can be realized in extended teleparallel gravity with a conformal scalar field, and that pure teleparallel gravity with a conformal scalar field can lead to the $\Lambda$CDM model. It is also shown that there can exist the de Sitter solution in conformal torsion gravity.Comment: 13 pages, no figure, version accepted for publication in Physical Review