How?

Are they actually sweating? There are systems for doing active cooling from fans running off battery power.

https://fursuitsupplies.com/fans

Or liquid cooling, including ice and evaporative cooling?

https://archive.ph/MiMVI

How a Cooling Vest Invented by a Furry Made Its Way Into the U.S. Military

Playing and performing in a full-body, mascot-like suit is a disciplined endeavor—and not always a comfortable one. Any furry will tell you that it’s hot inside a suit and that problem is compounded when members gather in real life. Fursuit-wearers run a real risk of overheating, especially at sunny outdoor events like meetups and Pride parades.

To increase their in-suit endurance, some furries use cooling vests, much like an athlete might. Specially designed vests can hold packs of fluids that remain at steady, low temperatures against wearers’ bodies—a godsend for keeping body heat at bay when you’re wearing fur head to toe.

Hmm. Also, while I’m no expert, my understanding is that animals that are in a cold climate, tend to have underfur to reduce convection. Like, dog breeds aimed more at warm climates don’t have that. I bet that one doesn’t need a lot of dense, short fur at the base. Even if there’s long fur, it might not be as insulating as one might expect relative to an animal.

considers

If the fabric isn’t actually visible — and for furred areas, I guess it isn’t, because all someone can see is fur — maybe one could use something like burlap at the base, stuff that has a lot of room for air to flow through.

I dunno what any issues with using carbon fiber are, but my understanding is that it’s pretty thermally-conductive. In some uses, carbon fiber is made into rigid surfaces, is a composite, has resin, but I believe that you can get it as a resin-less fabric.

kagis

https://www.ngfworld.com/en/fiber/high_thermal_conductivity.html

GRANOC high thermal conductive grade is available as Yarn, Fabric, Chopped and Milled fiber.

I mean, if you figure that it’s possible to use a fabric made out of that, that’ll probably conduct a lot of heat away from hot places.

https://pmc.ncbi.nlm.nih.gov/articles/PMC8867053/

According to this, carbon nanotube microfibers have thermal conductivity hundreds of times greater than nylon.

kagis

Apparently you get get fabric made from the stuff — though this example looks to be awfully expensive, so carbon nanontube-based fursuits probably aren’t practical for general use yet:

https://dexmat.com/store/galvorn-carbon-nanotube-fabric/

Galvorn carbon nanotube (CNT) fabric is highly conductive and made from interlocking loops of Galvorn CNT yarn.

Galvorn yarns behave like a textile. You can sew, weave, knit, and even blend it with other textiles to achieve your application goals. You can modify the yarn and/or knit to adjust the suppleness of the fabric.

https://dexmat.com/carbon-nanotube-fiber/

Thermal Conductivity: Galvorn CNT fiber has a thermal conductivity of 450 W/m-K, exceeding copper’s 385 W/m-K. This superior heat dissipation capability is vital for high-performance electronics and power cables, allowing for greater current carrying capacity per unit mass.

https://dexmat.com/industries/e-textiles/

I doubt that people are actually going all the way to carbon nanotube fabric, but if we can get costs down, I imagine that they could.