  Testing the constancy of the gravitational constant G has been a longstanding
fundamental question in natural science. As first suggested by Jofr\'{e},
Reisenegger and Fern\'{a}ndez [1], Dirac's hypothesis of a decreasing
gravitational constant $G$ with time due to the expansion of the Universe would
induce changes in the composition of neutron stars, causing dissipation and
internal heating. Eventually, neutron stars reach their quasi-stationary states
where cooling due to neutrino and photon emissions balances the internal
heating. The correlation of surface temperatures and radii of some old neutron
stars may thus carry useful information about the changing rate of G. Using the
density dependence of the nuclear symmetry energy constrained by recent
terrestrial laboratory data on isospin diffusion in heavy-ion reactions at
intermediate energies and the size of neutron skin in $^{208}Pb$ within the
gravitochemical heating formalism, we obtain an upper limit of the relative
changing rate of $|\dot{G}/G|\le4\times 10^{-12}yr^{-1}$ consistent with the
best available estimates in the literature.
