  A hypothetical time-variation of the gravitational constant $G$ would make
neutron stars expand or contract, so the matter in their interiors would depart
from beta equilibrium. This induces non-equilibrium weak reactions, which
release energy that is invested partly in neutrino emission and partly in
internal heating. Eventually, the star arrives at a stationary state in which
the temperature remains nearly constant, as the forcing through the change of
$G$ is balanced by the ongoing reactions. Using the surface temperature of the
nearest millisecond pulsar (PSR J0437$-$4715) inferred from ultraviolet
observations and results from theoretical modelling of the thermal evolution,
we estimate two upper limits for this variation: (1) $|\dot G/G| < 2 \times
10^{-10}\mathrm{yr}^{-1},$ if the fast, "direct Urca" reactions are allowed,
and (2) $|\dot G/G|<4\times 10^{-12}\mathrm{yr}^{-1},$ considering only the
slower, "modified Urca" reactions. The latter is among the most restrictive
upper limits obtained by other methods.
