Is there a good way to figure out whether my pet is running a temperature?

Use a thermometer.

The normal body temperature of a cat or dog at rest is 99 to 101¦F. If your pet is acting sluggish or depressed, is reluctant to eat or move around, is panting or breathing rapidly, or feels warm to the touch, it is exhibiting symptoms of a fever. The most accurate method of determining whether an animal has a fever is to use a rectal thermometer.

As with taking the temperature of an infant, the thermometer should be shaken, lubricated with petroleum jelly, and inserted about one-and-a-half inches into the rectum. It is helpful to have someone hold and distract your pet while its temperature is being taken.

If the reading is above 102.5¦F, you should contact your veterinarian. A temperature of 105¦F or more will require an emergency visit to a veterinarian. If the pet holds a temperature above 106¦F for very long, it could suffer serious adverse effects, including permanent brain damage and death.