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–Lemaˆıtre–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.

In the context of $f(R,T)$ theories of gravity, we study the evolution of scalar cosmological perturbations in the metric formalism. According to restrictions on the background evolution, a specific model within these theories is assumed in order to guarantee the standard continuity equation. Using a completely general procedure, we find the complete set of differential equations for the matter density perturbations. In the case of sub-Hubble modes, the density contrast evolution reduces to a second-order equation. We show that for well-motivated $f(R,T)$ Lagrangians the quasistatic approximation yields to very different results from the ones derived in the frame of the concordance $\Lambda \mathrm{CDM}$ model constraining severely the viability of such theories.

In the context of $f(R)$ theories of gravity, we address the problem of finding static and spherically symmetric black hole solutions. Several aspects of constant curvature solutions with and without electric charge are discussed. We also study the general case (without imposing constant curvature). Following a perturbative approach around the Einstein-Hilbert action, it is found that only solutions of the Schwarzschild-(anti) de Sitter type are present up to second order in perturbations. Explicit expressions for the effective cosmological constant are obtained in terms of the $f(R)$ function. Finally, we have considered the thermodynamics of black holes in anti-de Sitter space-time and found that this kind of solution can only exist provided the theory satisfies $R_0+f(R_0)<0$. Interestingly, this expression is related to the condition which guarantees the positivity of the effective Newton’s constant in this type of theories. In addition, it also ensures that the thermodynamical properties in $f(R)$ gravities are qualitatively similar to those of standard general relativity.

In this work we consider the possibility of describing the current evolution of the universe, without the introduction of any cosmological constant or dark energy (DE), by modifying the Einstein-Hilbert (EH) action. In the context of the $f(R)$ gravities within the metric formalism, we show that it is possible to find an action without cosmological constant which exactly reproduces the behavior of the EH action with cosmological constant. In addition the $f(R)$ action is analytical at the origin having Minkowski and Schwarzschild solutions as vacuum solutions. The found $f(R)$ action is highly nontrivial and must be written in terms of hypergeometric functions but, in spite of looking somewhat artificial, it shows that the cosmological constant, or more generally the DE, is not a logical necessity.

In the context of f ( R ) = R + α R 2 gravity, we study the existence of neutron and quark stars for various α with no intermediate approximation in the system of equations. Analysis shows that for positive α the scalar curvature does not drop to zero at the star surface (as in general relativity) but exponentially decreases with distance. Also the stellar mass bounded by star surface decreases when the value α increases. Nonetheless distant observers would observe a gravitational mass due to appearance of a so-called gravitational sphere around the star. The non-zero curvature contribution to the gravitational mass eventually is shown to compensate the stellar mass decrease for growing α ’s. We perform our analysis for several equations of state including purely hadronic configurations as well as hyperons and quark stars. In all cases, we assess that the relation between the parameter α and the gravitational mass weakly depends upon the chosen equation of state. Another interesting feature is the increase of the star radius in comparison with general relativity for stars with masses close to maximal, whereas for intermediate masses 1.4 – 1.6 M ⊙ the radius of star depends upon α very weakly. Also the decrease in the mass bounded by star surface may cause the surface redshift to decrease in R ² -gravity when compared to Einsteinian predictions. This effect is shown to hardly depend upon the observed gravitational mass. Finally, for negative values of α our analysis shows that outside the star the scalar curvature has damped oscillations but the contribution of the gravitational sphere into the gravitational mass increases indefinitely with radial distance putting into question the very existence of such relativistic stars.