Edit: A comment pointed out that I didn't answer the question. Dry cotton is a good insulator, and most cotton t-shirts are thicker on average than synthetic shirts. A typical cotton tee in dry weather will be warmer than a typical dry synthetic shirt.
Now start to work hard. The cotton takes a fair amount to soak through. So even after you start to sweat, it takes a while before evaporation is happening on the surface of the cotton. This cooling effect will continue after you've stopped working, as you have a considerable backlog of sweat in the shirt.
This has happened to me many times: I chug up the last thousand feet of scree to the pass sweating in the sun. Step, step breath. step step breath. Thinking, "I'm too old for this stuff." I get to the top. Of course there's a wind. Yes my synthetic shirt has some sweat in it, especially where my pack and straps sat.
I'm chilled. But not as much as I am with cotton. The greater porosity of the the synthetic meant I had a fair amount of direct evaporation. What sweat that's in the shirt moved to the surface (wicking...) and dried there. So I got more benefit right away when I started my upward trudge.
TL;DR: Because of cotton's water storage there is a bigger lag during variable activity. You can be too hot and too cold at various points in your activity cycle.
There is a reason people wear cotton in tropical climates. And on my canoe trips in summer, one item is a light weight light coloured (not white. beige or light green) long sleeve cotton shirt, as the best protection against bugs.
The problem with cotton is that the fibres soak up water and it takes a long time to dry. During that long time, the water evaporates, cooling you.
Scenario: You're on a hike in jeans and t-shirt. Squall line comes through and drops a quarter inch of rain in 20 minutes, followed by a cold front. The day drops from 70 F to 50 F, with a stiff wind. That wind evaporates the water in your clothes putting a layer of wet cloth at about 40 F next to your skin. Some of that is balanced by bright sun, but you are still cold, and working your way toward hypothermia
Change it to nylon pants and a polyester shirt. The pants don't bind as much, so your skin isn't next to a layer of wet fabric. A lot of that water drained through the fabric down to the cuffs and dripped off. The shirt because of it's fine weave (that's how they wick) will hold water on the fabric, but not in it. Take it off, wring it out put it on damp.
Net result: A lot less chilling.
Scenario: It's been drizzly all day. You are moderately soaked. Come into camp, build a fire. With wet jeans, you are steaming for an hour. With nylon wind pants you are steaming for 10 minutes.
Scenario: You're hiking and cross a stream. You lose your footing, and go SPLASH!. In cotton you are now carrying 5-10 pounds more weight that will gradually reduce over the next two hours. With synthetic you are carrying 1-2 pounds more weight that is dry to touch in 20 minutes to half an hour.
To me the problem of cotton isn't that it's 'cold' but that it stays cold and wet for long periods of time. In hot weather this is a win. Take off your shirt, soak it in the creek, pull it back on (gasp as cold shirt hits warm ribs...) and you are cooled for a while from the shirt's evaporation.
When running canoe expeditions I recommended two cotton tee shirts, and one synthetic. If the weather looked to be cool, or wet, or I knew that the day would include lots of in and out of the canoes tracking and lining, I would recommend to the kids that they wear their synthetic.
It doesn't have to be expensive. I have 5 drywick synthetic shirts now. Got them all at goodwill, paying about a buck more than a cotton tee would have been. One has a corporated logo, one has Carnival's logo. Nylon pants in the sports wear. I prefer to get unlined ones, as they are more versatile, but I have a couple pair with poly mesh lining that I use when working outside in winter.
I got one goretex Sierra Designs jacket at Value Village for $8. The liner has failed somewhat. Not longer fully waterproof, but I use it when I doing errands on rainy days.
There are certain other circumstances that cotton is fine in addition to the hot weather scenario. I pick up a couple of 'quilties' each fall. Cotton flannel outside, quilted dacron fiber inside. This is a nice combination when I'm working outside on my farm. Nice balance of wind resistance and warmth. I've used them on day trips snowshoeing. They are cheap, but they tear on sharp wood ends, and barbed wire fences. By the end of a year or two they are dog bedding.
But I would never take them on an overnight. Get them wet and they are a real challenge to get dry. At home I either hang them on a hook by the wood stove, or chuck them in the dryer.
Cotton gloves are the worst: A glove or mitten is almost always wet by the end of the day, or at least damp. It is very difficult to get the right balance of warmth and wind resistance. Hands go through a cycle of flushing with blood when you are cold, so their temp cycles too. When they are warm, they sweat, especially the palms. (Trick: apply antipersperant to your hands and soles of your feet to reduce the sweating and chilling issue)