From Longman Dictionary of Contemporary Englishrun a temperature/feverrun a temperature/feverMIto have a body temperature that is higher than normal, because you are ill → run
Examples from the Corpus
run a temperature/fever• By Sunday morning she was running a temperature.• Lice are very temperature-sensitive and will leave a body which is running a fever.• Lips all cracked, glassy eyes, running a temperature.• No wonder she was running a temperature!• She wasn't tipsy, just a little high, probably running a fever.• I think he's running a temperature, and he's off his food.• Not running a temperature, are you?• Next morning he ran a temperature but insisted on getting up.• The baby was fussing and running a fever, so I called the doctor. 