To despond is to become very downhearted or gloomy. You could say that you tend to despond whenever you think about the end of summer.
As a verb, despond is an old-fashioned word that you'll probably only see in books. It's much more common to hear the related words despondent and despondence. Still, if you don't mind sounding a little outdated, you can urge your little brother not to despond over his glass of spilled milk. The Latin root is despondere, "to give up, lose, lose heart, or resign."