Should I give my router a “break” between passes?

I normally always switch off all my power tools in between setup and next pass especially when working with large pieces. The power switch is located and convenient to access. So turning off the tool is more for safety. All prep work would mean the power tool is turned off.

With a router underneath the router table, without a separate convenient switch location, this may not be easily done.

I am building a router table now, that will have an easy-to-reach push cutoff switch.

To answer your specific question about cooling, you should not be feeding the router that hard that it requires a cooling-off period. If you see that the router is starting to heat up, then a cooling off period is needed. Your router may cool off faster with the router on with the fan blowing air through the motor.

The motor/bearings in the router aren't generally much of a concern for overheating. However, router bits can definitely overheat. Particularly, if you're using a high speed steel (HSS) bit overheating it will cause it to loose hardness and dull very quickly.

Carbide is generally able to dissipate heat better than HSS, but if you get them too hot they will be more prone to chipping.