Entropy relations and bounds of horizons in modified gravity

We first give some entropy relations for black holes in modified gravity models, which are introduced as an elegant technique trick for handling various entropy bounds. For these entropy relations, some are mass-independent and universal, while others are not. Then we study the entropy bound of horizons in modified gravity, including Horava-Lifshitz gravity, massive gravity and Einstein-dilaton gravity. We focus on black holes with two or three physical horizons, containing the event horizon, Cauchy horizon and negative horizon which is physical for the observers in the negative radial coordinate region. In these modified gravity, entropy bounds are geometrical bounds which are related close to the cosmological radius for asymptotically (A)dS black holes, while they are Penrose-like inequalities for asymptotically flat black holes. Moreover, they depend on the constants characterizing the strength of modified terms in the actions. We also present the first law of thermodynamics and Smarr relations for horizons