Yes, you'll want some cooling. When they talk about the recommended temperature, that means the temperature of the fermenting beer, not the ambient temperature. Fermentation generates heat (same way mulch piles and garbage dumps catch fire,) and can raise the fermentation temperature by 8-10°F above ambient at peak activity. Ynotbrusum suggests a very simple method which will serve you well. Some people cover the fermentor with a wet towel or t shirt that reaches into the water to also get some evaporative cooling, but in a closet with no moving air, or a humid environment, that won't give you much of an advantage.
Good luck, and welcome to the hobby!