  Here we apply the Generalized Second Law of Thermodynamics (GSL) to black
holes accreting and emitting in the present Universe and derive upper limits on
the variation in the gravitational constant G. The limits depend on how the
gravitational mass M varies with G. Parameterizing M goes as G^n, if n > -1/2
(including n = 0), the GSL applied to the full range of black holes
theoretically allowed in the present Universe does not constrain an increase in
G but any decrease must be less than about |(1/G) dG/dt| = 10^-52 per second.
If n < -1/2, the GSL does not constrain a decrease in G but any increase must
be less than about |(1/G) dG/dt| = 10^-52 per second. At earlier redshifts,
these constraints weaken as z^3. If n = -1/2, the GSL does not constrain a
decrease but any increase must be less than about |(1/G) dG/dt| = (1/t). If the
mass range is restricted to those black holes which have been astronomically
observed, the present constraints on n > -1/2 and n < -1/2 are only weakened by
a factor of about 10^8 with the tightest constraints coming from stellar mass
black holes and the n = -1/2 bound does not change. The stellar mass black hole
limits should constrain the variation of G in Standard Model physics and all
extension models which approximate classical physics on astronomical scales.
