Molar gas constant is the same for all gases because at the same temperature and pressure, equal volumes of all gases have the same 

$\text{ We know, }\mathrm{R}=\frac{\mathrm{PV}}{\mathrm{nT}}$

assuming P,V and T to be constant.
Also we know, that equal volume of gases contain equal number of molecules, hence the value of R is also constant.