Comment by yamrzou

If you want an independent definition of temperature without reference to entropy, you might be interested in the Zeroth Law of Thermodynamics (https://en.wikipedia.org/wiki/Zeroth_law_of_thermodynamics).

Here is a intuitive explanation for it from [1]:

“Temperature stems from the observation that if you bring physical objects (and liquids, gases, etc.) in contact with each other, heat (i.e., molecular kinetic energy) can flow between them. You can order all objects such that:

- If Object A is ordered higher than Object B, heat will flow from A to B.

- If Object A is ordered the same as Object B, they are in thermal equilibrium: No heat flows between them.

Now, the position in such an order can be naturally quantified with a number, i.e., you can assign numbers to objects such that:

- If Object A is ordered higher than Object B, i.e., heat will flow from A to B, then the number assigned to A is higher than the number assigned to B.

- If Object A is ordered the same as Object B, i.e., they are in thermal equilibrium, then they will have the same number.

This number is temperature.”

[1] https://physics.stackexchange.com/a/727798/36360