There are two jumps in your logic here; First, that heat=energy. While it's simplified for most intro physics courses, heat=/=energy. Second, that the brain using energy produces heat, like a muscle. A muscle gets hot when it works due to friction and breakdown as the muscle fibers "scratch" and tear past each other.

This damage can also cause inflammation in the area (blood rushes in) to help it heal faster. This is the same reason your skin feels hot when you have a sunburn. (The feeling is due to more blood in the inflamed areas. These areas are close to neurons that feel temperature, so the neurons think (perceive) you're hotter than normal when it's really just a part of your body abnormally hot, not your whole body.)

Blood provides neurons with "raw materials" for energy (sugar and oxygen). When a brain region is active, there is hypothesized to be an increase in blood flow to that region, helping those neurons to keep producing energy. This is the basis of BOLD analysis of fMRIs (Blood Oxygen Level Dependent analysis).

The brain is always working hard, and is actually hotter than your core. 30% of your caloric expenditure is because of your brain. This is why cooking is so important; it pre-digests food, so our body spends less energy digesting, and instead gets more free calories from food. But that's a talk for a different time.

When neurons are active, they don't move, unlike muscle cells. Since there's no moving, there's really no change in heat. There are no heat sensing neurons in your brain so you wouldn't sense a perceived change in heat (unlike the sunburn scenario). There can be an increase in temperature in spots in the brain, but that's due to inflammation. Here's a pubmed article that goes into brain temperature regulation a little.

TLDR; Energy use =/= an increase in heat. Your neurons might use energy, but they don't produce heat. There isn't a need for a cooling system unless there's a systemic problem, like a fever.

BONUS: Who told you it cools itself through the ears?? There's not brain matter under your ear drums. It's your tympanic membrane, cochlea, and oodles of more little bones and pieces.