  We formulate the basic framework of thermodynamical entropic force cosmology
which allows variation of the gravitational constant $G$ and the speed of light
$c$. Three different approaches to the formulation of the field equations are
presented. Some cosmological solutions for each framework are given and one of
them is tested against combined observational data (supernovae, BAO, and CMB).
From the fit of the data it is found that the Hawking temperature numerical
coefficient $\gamma$ is two to four orders of magnitude less than usually
assumed on the geometrical ground theoretical value of $O(1)$ and that it is
also compatible with zero. Besides, in the entropic scenario we observationally
test that the fit of the data is allowed for the speed of light $c$ growing and
the gravitational constant $G$ diminishing during the evolution of the
universe. We also obtain a bound on the variation of $c$ to be $\Delta c/c
\propto 10^{-5} >0$ which is at least one order of magnitude weaker than the
quasar spectra observational bound.
