The universe was at its hottest when it was at its smallest. As the universe expands, it gets cooler.
Is the expansion of the universe the direct cause the cooling?
I suppose it would be more clear to me if someone defined the meaning of 'temperature'.
The expansion of the universe is related to its cooling.
Temperature is defined as the inverse of the partial derivative of entropy with respect to energy: $$ \frac{1}{T}=\frac{\partial S}{\partial E} $$
The derivative is taken at constant volume. This is the definition of temperature. This definition only makes sense for macroscopic bodies. I.e., temperature only makes sense for such bodies.
