  Observational evidence suggests that gas disk instability may be responsible
for the formation of at least some gas giant exoplanets, particularly massive
or distant gas giants. With regard to close-in gas giants, Boss (2017) used the
$\beta$ cooling approximation to calculate hydrodynamical models of inner gas
disk instability, finding that provided disks with low values of the initial
minimum Toomre stability parameter (i.e., $Q_i < 2$ inside 20 au) form,
fragmentation into self-gravitating clumps could occur even for $\beta$ as high
as 100 (i.e., extremely slow cooling). Those results implied that the evolution
of disks toward low $Q_i$ must be taken into account. This paper presents such
models: initial disk masses of 0.091 $M_\odot$ extending from 4 to 20 au around
a 1 $M_\odot$ protostar, with a range (1 to 100) of $\beta$ cooling parameters,
the same as in Boss (2017), but with all the disks starting with $Q_i = 2.7$,
i.e., gravitationally stable, and allowed to cool from their initial outer disk
temperature of 180 K to as low as 40 K. All the disks eventually fragment into
at least one dense clump. The clumps were again replaced by virtual
protoplanets (VPs) and the masses and orbits of the resulting ensemble of VPs
compare favorably with those of Boss (2017), supporting the claim that disk
instability can form gas giants rapidly inside 20 au, provided that
sufficiently massive protoplanetary disks exist.
