I posted before with a PATS problem with my 2006 Mustang.  I mentioned
that I had put a Saleen supercharger in the car and that the dealer
was claiming that is what was causing my PATS problem

Update - I spent 5 days experimenting with the car in the Florida
summer heat.  I found that the PATS issue was indeed heat related - I
could reproduce the PATS lockout by parking the car in a certain spot
in the driveway, facing east, and by about 12:00 pm or 1:00 pm the car
would not start due to PATS.  I kept a thermometer in the car and the
inside temperature was approximately 110 - 150 degrees F.  I
experimented with several heat-reducing ideas (such as a sunshade).
While the sunshade did reduce the heat, it did not reduce it enough to
keep the interior temperature from reaching 120 F.

Each day, when the internal temperature would reach 110 - 120 F, the
car would be locked out by PATS and would not start.  Then, in the
evening and doing nothing more to fix the car than to wait until it
cooled down, the car would start fine.

So, on the 5th day I decided to put the car back in the same spot and
wait until PATS failed.  It failed again by 1:00 pm.  This time I
tried disconnecting an aftermarket AFR gauge I had installed, in case
my choice of circuits that I tied into somehow impacted the PATS
circuit.  I disconnected the AFR gauge and still the car would not
start.

I then proceeded to remove the plastic covers (upper and lower) that
surround the steering column to look at the PATS receiver.  It turns
out to be a very simple device.  It is basically a plastic case that
surrounds where the key goes (which is the copper antenna) and a small
circuit board.  A four-wire quick disconnect connector is the only
connection.  With the cover off I tried to start the car - nothing.  I
then simply wiggled the wire harness that connects to the PATS
receiver.  The car started right up.  I removed the PATS receiver from
the car and split the plastic case open to look at the internal
circuitry.  A small circuit board with a few chip components connected
to a copper wire coil antenna.

I put the PATS cases back together and took it back out to the car.  I
connected it to the wire harness but did not put it back into position
around the key.  I wanted to see if it needed to be surrounding the
ignition or if it could read the key in any position.  With the PATS
dangling by its connection I inserted the key and brought the PATS
antenna close to the key fob (which is most likely where the passive
RFID chip is located).  The car would not start.  I put the PATS
receiver back in its proper position and the car started.  Seems the
PATS receiver must be in its proper position surrounding the ignition
for it to work.

I did notice a rubber grommet which did not appear to be properly
installed by virtue of the fact that one side of it had been
improperly pressed together.  I spent some time installing the rubber
grommet directly around the plastic PATS antenna.  With the grommet
installed the PATS receiver fit much more snuggly into position than
before.  I suspect the grommet acts as a shock isolator for PATS as
well as a thermal insulator.  If so, it had not been properly
installed.  Time will tell.

Anyway, the car starts fine now and it is now a matter of time (and no
more PATS lockouts) before I can feel confident that I solved the
problem.

As I still suspect, this issue has nothing to do with the supercharger.