Is there a thermostat that can be replaced? Yes, of course.

Will it fix your problem? No.

Sounds to me like your compressor is shutting off on the hot days due to
high head pressure.  This means that the output pressure from the compressor
is too high, there is a pressure switch in the system that monitors the head
pressure and shuts down the compressor if it exceeds a preset limit.  The
causes for this can be a defective fan circuit, dirty condensor (radiator),
a restriction in the compressor discharge line, too much charge, or air in
the system.  First thing to check  is to see if your auxilliary fan kicks on
when you turn the AC on ... lift the hood, turn on and off the AC, a second
fan should turn on and off right away with the AC dash switch.  If this
doesn't happen, the problem is in the fan electrical circuit; a defective
switch, relay, fan motor, or connection.   Not knowing how your vehicle is
set up, I'm offering generic instructions .. your vehicle might not have all
this equipment.

Next, take a hose with a kind of high pressure, not really high, and hose
down the radiators on your car, thoroughly .. from a straight on angle,
perpendicular to the coils.  If you live somewhere where the are lots of
Dandylions, of Cottonwood trees, or your grill is packed with dead bugs, a
regular flushing of the radiators should be performed.  You could damage the
coils if the pressure is too high.

If these actions don't cure the problem, then take it to a service tech and
have them evacuate and recharge the system again .. request that they are
careful about the amount of charge they put in.  Try not to insult them, so
maybe take it to a different tech.

Most likely, one of these things will solve your problem.  The first two are
the easiest to check and least expensive to fix.  You know about the cost of
the thrid.

Hope this helps.