The term “hot seat” stems from use of the electric chair in executions, a practice started in the late 19th century in that doomed NFL city of Buffalo, NY. The convicted party is strapped to a chair with electric currents running through it. The first jolt stops the heart, causing unconsciousness. The rest of the electricity completes the kill.
One can imagine getting strapped into the chair and anticipating its cruel heat before any electricity has been summoned. Though the method of execution is now nearly extinct, the expression “hot seat” has life, as it were, in the world of coaching.