Once we define the entropy its integrating factor is what we conventionally call temperature. I agree that correspondence between things is not obvious - but on

Once we define the entropy its integrating factor is what we conventionally call temperature. I agree that correspondence between things is not obvious - but once defined it seems to work and I don’t know if we could have more proof than that. I also agree that thermodynamics is worth studying in itself - and otherwise the attempt to reproduce it using statistical mechanics doesn’t make a lot of sense.