Not really, only coil size does.
If you keep the diameter and length of the coil exactly the same; it does not matter which gauge you use; the coil will always have the same (or very close to it) heat flux.
Easy example would be, lets say 0,25 mm vs 0,5 mm. You wrap a contact coil of the same size with both wires. You´ll need exactly double the wraps from the 0,25mm wire compared to the 0,5mm wire to achieve this.
Heat flux is calculated by total surface area. Surface area of the total coil wire is calculated by π * wire diameter * lengths of wire. π will, of course, always stay the same.
So lets say you use 20cm of the 0,25mm wire and 10cm of the 0,5mm; you´ll get π * 0,25 * 20 vs π * 0,5 * 10.
Its the same.