Power law cosmology in modified theory with thermodynamics analysis

In this paper, we consider a cosmological model in $f(R, G)$ gravity in a flat space-time, where $R$ is the Ricci scalar and $G$ is the Gauss-Bonnet invariant. The function $f(R, G)$ is taken as a linear combination of $R$ and an exponential function of $G$. We analyze the observational constraints under a power law cosmology which depends on two physical parameters: the Hubble constant $H_0$ and the deceleration parameter $q$. We constrain these two dependent parameters using the latest 77 points of the OHD data, 1048 points of the Pantheon data, and the joint data OHD+Pantheon and compare the results with the $\Lambda$CDM. Also, we speculate constraints using a simulated data set for the future JDEM (Joint Dark Energy Mission)/Omega, supernovae survey. We see that $H_0$ is in very close agreement with some of the latest results from the Planck Collaboration that assume the $\Lambda$CDM model. Our work in power law cosmology better fits the Pantheon data than the earlier analysis \cite{Kumar:2011sw, Rani:2014sia}. However, the constraints obtained on $H$ average, $<H_0>$ and $q$ average, $<q>$ using the simulated data set for the future JDEM/Omega, supernovae survey are found to be inconsistent with the values obtained from the OHD and the Pantheon data. Additionally, we discuss statefinder diagnostics and see that the power law models approach the standard $\Lambda$CDM model ($q\rightarrow -1$). This model satisfies the Generalized Second Law of Thermodynamics. Finally, we conclude that the power law cosmology in $f(R, G)$ gravity explains most of the distinguished attributes of evolution in cosmology.