Relationship between the ideal gas constant and entropy
The gas constant is equal to Avogadro's constant times Boltzmann's constant, the latter serving as a proportionality constant between the average thermal (kinetic) energy of the particles in an ideal gas and the temperature:

$$\left(\frac{\partial \bar U}{\partial T}\right)_p=\frac{3}{2}k_\mathrm{B}$$

The entropy can be regarded as a proportionality constant between the change in free energy $G$ with change in temperature of a system at constant pressure, since

$$\left(\frac{\partial G}{\partial T}\right)_p=-S$$

Note also that the average entropy of a particle can be written as

$$S=k_\mathrm{B} \log\Omega$$

So both $R$ and $S$ can be regarded as proportionality constants between energy and temperature.

The Wikipedia page on the equipartition theorem may provide some enlightenment on the origin of these proportionalities.