If you want to see what barrel temperature does to your POI, get a package of temperature tape strips - I use the 85 to 140 deg. F. versions now but started with 85 to 190 deg. F. versions but never allowed a barrel to get hot enough to use the higher portion of the temperature range.
Then to understand the effect of temperature on POI you have to do some 'heavy duty' data gathering and perform some statistical analysis.
You'll need a pretty large sample to allow conclusions since you will most likely have a contribution from 'shooter induced variations' that will effect the results.

I won't claim that the temperature strips are 'calibrated' but the measurements appear to be consistent.
I did verify the readouts with and inferred temperature gun when I first got them. The tapes might have been off by 2 degrees compared to the inferred temp gun.

I have found that I begin to see some POI movement above 115 to 120 deg. F. readings on the tape.
My most accurate rifles seem to drop the POI about 1/4 inch at max at around 130 deg. F. but the groups are still small - the POI is just below the POI.

I might see group size growth if some were shots were in a cold barrel and some were shot from a hot barrel, but I don't shoot groups that way.
In fact, I don't know how you might get 40 degrees increase in temperature when shooting a 5 round group.
On a hot day it takes about 12 rounds of 6.5mm CM to raise the temperature of the barrel from 85 deg. to 115 deg. F.
My .223 bolt actions take about 18 to 20 rounds to get the barrel above 115 deg. F but the cartridges are loaded with 15 to 18 grains less powder per round.
We have a covered firing line so the rounds are out of the sun.

I would think that heating up your ammo in the sun with a temperature insensitive powder could cause more temperature variation effects than a hot barrel as long as you aren't allowing your rounds to cook in a hot barrel before firing them.