The type of biome you get depends largely on availability of water, not temperature.

Deserts are deserts because they have very poor availability of water most of the time. This is most often caused by simple lack of precipitation, but other factors can influence this:

• High temperatures cause high evaporation rates, meaning to need more precipitation to achieve the same level of plant growth. This is why, for example, 10 inches (25.4 cm) of precipitation will get you desert in the tropics, subtropics, and temperate latitudes, but it’ll get you boreal forest in the colder subpolar latitudes.
• Extremely low temperatures (such as in Antarctica) result in everything being perpetually frozen. Most of Antarctica is a desert, both because it gets very little precipitation and because all the ice on the ground isn’t available as liquid water.
• Extremely sandy or gravelly soils which do not retain water cause poor water availability, even with abundant precipitation and a mild climate. While these aren’t typically classified as "true deserts), the plant life certainly reflects the harsh conditions and poor availability of water.

As for why we largely don’t see desert at the equator, it’s because of precipitation. Due to the circulation of cold and warm air in the atmosphere, the equator typically sees warm air, often laden with moisture due to the oceans and the high moisture capacity of warm air, rise. As it rises, it cools, and because cool air cannot hold as much moisture as warm air can, it drops a lot of that moisture as rain. This results in most of the equator getting a lot of rain.

Once the air has risen and cooled, it cycles north and south into the subtropics, where it falls down to earth again. And in falling, it warms up again, especially as these regions still receive a ton of sunlight, particularly in the summer. But the air has already lost much of its moisture, so now it’s just a bunch of hot, dry air blasting down over the subtropics. This is why we have bands of deserts across most of the subtropics, from the Sahara to the Middle East to the desert of the SW US and northern Mexico. Same on the opposite side of the equator, with the Kalahari desert in southern Africa, the Australian outback, and the Patagonian desert.

There are other factors, too, of course, such as rain shadows from mountains and ocean currents, but the atmospheric circulation is the big one to answer your question.