 This paper reviews various aspects of modified gravity theories, focusing on black hole solutions and their comparison with those in general relativity, as well as Friedman-LeMetro-Robertson Walker, FLRW, Metrics. It also examines the thermodynamic properties of fourth order gravity theories, paying particular attention to the local and global stability of F-R-H models. Additionally, the authors explore the extension of the Cardiol and formula to modified gravity, and investigate the validity of entropy bounds in such scenarios. Finally, the paper studies the appearance of cosmological singularities, which can occur in certain modified gravity theories, and assesses whether or not these singularities violate the entropy bounds. This article was authored by Diego César Gomez and Alvaro de la Cruz Dumbres.