Astrophys Space Sci (2014) 354:507–515 DOI 10.1007/s10509-014-2124-3 ORIGINAL ARTICLE Thermodynamics with corrected entropies in f (G) gravity M. Sharif · H. Ismat Fatima Received: 11 August 2014 / Accepted: 6 September 2014 / Published online: 13 September 2014 © Springer Science+Business Media Dordrecht 2014 Abstract We study generalized second law of thermody- namics for the flat FRW universe model in modified Gauss- Bonnet gravity. For this purpose, we assume scale factor in power-law form to construct f (G) model and check the validity of this law. We use power-law and logarithmic en- tropies with and without quantum corrections to the horizon entropy and take Hubble and event horizons as boundary of the universe. The graphical representation shows that this law holds for both present and future epochs but violates in the past era for power-law entropy with and without quan- tum corrections, while this law is valid for the whole range of redshift z for logarithmic corrected entropy. Keywords Generalized second law of thermodynamics · Entropy corrections · f (G) gravity 1 Introduction The idea to unify quantum mechanics and general relativ- ity (GR) is one of the main targets of modern physics. The attempts for unification of these two theories is named as quantum gravity which is not yet successful. Alternatively, various theories like string theory and loop quantum grav- ity provide information about new features of black holes and singularities. According to these theories, black hole M. Sharif (B ) Department of Mathematics, University of the Punjab, Quaid-e-Azam Campus, Lahore 54590, Pakistan e-mail: msharif.math@pu.edu.pk H. Ismat Fatima Department of Mathematics, Queen Mary College, Lahore 54000, Pakistan e-mail: ismatfatima4@gmail.com acts similar to a thermodynamical system with temperature T proportional to its surface gravity κ at the black hole hori- zon, i.e., T ∝ κ providing a connection between tempera- ture of thermodynamical system and surface gravity of black hole (Mignemi and Stewart 1993) as given by Hawking (1975). Bekenstein (1973) related entropy of horizon S and area of black hole A as S = A 4 , i.e., he found that if general- ized second law of thermodynamics (GSLT) is not violated in the presence of black hole, then it must possess an entropy proportional to the area of its horizon. Relativistic thermo- dynamics is very important for relativists to discuss analo- gies with black hole thermodynamics. Gibbons and Hawk- ing (1977) discussed the connection between gravitation and black hole thermodynamics. Jacobson (1995) derived the field equations by using Clausius relation TdS = δQ in the Rindler spacetime, where δQ is the heat flux through hori- zon. The link between thermodynamics and gravity has been confessed in modified theories of gravity. These theories of gravity (Kung 1995; Nojiri and Odintsov 2004, 2005; Gube- rina et al. 2006; Sadjadi and Honardoost 2007; Santos et al. 2007) attracted many people to explain different phenom- ena of the universe by modifying the Einstein-Hilbert ac- tion. The modified Gauss-Bonnet theory of gravity or f (G) gravity is obtained by adding an arbitrary function of G in the Einstein-Hilbert action (Nojiri et al. 2006), where G is the Gauss-Bonnet quadratic invariant. The motivation of this modification comes from string theory by low energy effec- tive scale (Cognola et al. 2006, 2007). This theory has been analyzed for the cosmic expansion of the universe (Easson 2005) to avoid four types of finite future singularities (No- jiri et al. 2005; Carter and Neupane 2006; Koivisto and Mota 2007), solar system tests (Nojiri et al. 2007), thermodynam- ics (Sadjadi 2011; Chatterjee and Parikh 2013) and many other phenomena.