My initial testing with a makeshift system included a gel ice pack on my front chest in the centre & another on my back in the centre, with a gel ice pack in each front leg pocket. These gel packs lasted about 90mins and I swapped them out for fresh ones after lunchtime break. They made life a LOT more bearable indeed.
I have only just finally finished my personal spec cool vest. I bought two breathable adult football bibs (like a side-less tabard vest) & cut one short to fit under the flack vest. I then used the other bib as a breathable material donor to sew on the pockets for the ice/gel packs to slot into. The jetpack harness goes on top of the vest/gels etc, then the flack vest, then the rear armour plate. It is a snug fit - I was worried about the added girth with these but I'm ok.
I now have two larger ice packs that go across my back - these are around 3/4" thick & contain water, then two tall thin gel filled packs that are 1/4" thick at the front. I may use a half height gel pack on the heart side of my chest as I read the ice pack may cause angina & blood vessel operation side effects.
I also have two smaller square water based ice packs that go into my leg map pockets. The water based ice packs last quite a bit longer than the gel type (just as advertised).
I checked into devising a Thermo Electric Cooler set up. The power requirements are too high to make it practical. You couldn't carry a battery pack and conceal it in your Jet Pack or anywhere else big enough to provide cooling for any length of time.