  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.
