[Rayhem]

> Also I wonder if there's an intensive thermodynamic property that actually says how much thermal energy is in the system, since temperature apparently won't do it?

Thermodynamic beta[1] does exactly that. If we consider temperature as "tendency to give energy away", then the scale starts at zero, heads out through positive infinity, comes in through negative infinity, and then stops at negative zero. I.e. if T_a > T_b > 0, then system a gives energy to system b. Then, if T_c < T_d < 0, then system d gives energy to system c.

Thermodynamic beta (really just 1/T, the "coldness" of a system) fixes this: if B_a < B_b anywhere on the number line, then B_b is colder than B_a.

[1]: https://en.wikipedia.org/wiki/Thermodynamic_beta