T dS = δq, or dS = δq / T. The product of temperature and entropy change is energy change. If I use units in which Boltzmann’s constant k is unity and dimensionless, then entropy change dS is dimensionless, and temperature T is measured in the same units as δq, i.e., kelvins. Then T is a very small number, because the temperature is essentially one-half the average energy of a degree of freedom of a system, and any macroscopic sized system has on the order of 1023 degrees of freedom.

So, the question of “why is hot energy more valuable than cold energy?” seems to lead to the answer, hot energy is contained in fewer degrees of freedom than cold energy.