Cold packs can be a good way to cool down, but there's two downsides. Many cold packs start a chemical process that is hard to stop, meaning it'll continue until it's finished. The second is that it might cool down too fast, chilling the creature and reducing the muscles and brains effectiveness. This could even be dangerous.
You need a well controlled way to cool down. For that the cheetah can use blood fur.
Blood fur are tiny but tough hair follicles that have two hollow tubes inside them. They are connected to the blood vessels on the skin, allowing blood to flow to and from the hair follicles. In the skin each blood vessel to a follicle can be constricted individually, which is triggered by damage to the follicle. When it gets damaged it'll release a tiny hormone that shuts it down. In addition, the nervous system can autonomously change tiny regions in the skin on or off. This way the blood flow to the follicles is controlled, making sure blood only goes there when cooling is needed. It'll also allow only parts of the body to cool down, controlling how much heat is expelled. This os much like the sweating system of humans, only with blood through hairs.
Using blood in hairs will increase the surface area a thousandfold. As blood is the main transporter of heat, the closeness to the air inside a hair will allow heat exchange at a great level. Cooling down can be done at a very precise level, preventing over or undercooling. The fur can also become very thick for further protection, as cooling is now not a problem anymore.
In addition, when the cheetah attacks and starts using the blood fur to cool, it'll change to a blood red colour in places.