Recently it has shown that Einstein's field equations can be rewritten into a form of the first law of thermodynamics, which is indicates intrinsic thermodynamics properties of spacetime horizon. In the present thesis we deal with the so called f(R) gravity, whose action is a function of the curvature scalar R. In the set up of static spherically symmetric black hole spacetime, we find that at the event horizon, the field equation of f(R) gravity can be written into a form dE=TdS-PdV+TdS' where T is the Hawking temperature, V is the volume of the black hole horizon and dS' can be interpreted as the entropy production term due to non equilibrium thermodynamics of spacetime. Compared to the case of Einstein's general relativity, an additional term dS' also appears here. The appearance of the additional term is consistent with the argument recently given by Eling et al that the horizon thermodynamics in non equilibrium one for f(R) gravity. By applying the first law of thermodynamics dE=TdS to apparent horizon of the (n+1)-dimensional FRW universe and by working out the heat flow through the apparent horizon, we are able to drive the Friedmann equations of the FRW universe with any spatial curvature. Also by using the entropy expression of a static spherically symmetric black hole in the Gauss-Bonnet gravity and in more general Lovelock gravity, the corresponding Friedmann equations in each gravity can be reproduced. The fact that the higher order corrections to the WKB ansatz yields quantum corrections to the black hole temperature enable us to find the entropy corresponding to the modified temperature. In this thesis, we study the thermodynamics of the third order Lovelock black hole and BHT black hole. Then calculate the correct Howking-like temperature and after that the corrected entropy is computed for these kinds of black holes. It is shown that Lovelock black hole Howking radiate with a perfect black body spectrum and due to quantum modified thermal nature of the spectrum does not change. The most important property of Lovelock black hole is that it's corrections to the semi Key Words: Black Hole Thermodynamics, f(R) Gravity, Lovelock Gravity, Quantum Tunneling , Massive Gravity, BHT Black Hole.
