Before zeroth law of thermodynamics In my textbook and other places on internet, it is stated that zeroth law of thermodynamics is called so because it was formulated long after the other three laws were already in practice. Also, according to my understanding of the zeroth law, it establishes temperature as a state function.
But well, I think that there are many concepts based on the other three laws where we use that temperature is a state function. For eg., we say that change in internal energy of an ideal gas is dependent only on temperature, but well if temperature is not a state function then $U$ can also not be stated as a state function.
So how did scientists work out all these things before they established the zeroth law? Or is it that it was actually accepted that temperature is a state function but they just took it for granted until someone pointed it out? Also, if we have a law for temperature, why leave pressure and volume out? I mean why don't we get a law for establishing them as a state function as well?
 A: There is a fundamental difference between temperature we knew before statistical mechanics and after Boltzmann. Before Boltzmann, temperature was a relative measure for the warmness of matters.
But Boltzmann (1875) defined entropy as $S(E)=k_B \ln W(E)$ and as its consequence $\frac{1}{T}=\frac{\partial S(E)}{\partial E}$ where $T$ here, is an absolute value. It is amazingly explained in Schwabl G, Statistical Mechanics, section 2.4.1 (Definition of Temperature).
Of course at some limit these two values converge, but generaly one should have in mind the difference and yet, it took some years for scientists to build a bridge between these concepts.
Lastly, the state functions of a systems depends on which ensemble approach we take in statistical mechanics and sure, volume and pressure can also be state functions (List of state funcions).
A: The zeroth law of thermodynamics has a strange status among the fundamental laws of this subject. Its statement is so simple that there are not many variations of its formulation. It can be seen as postulating the existence of a scalar quantity (the temperature) that uniquely characterizes the mutual thermal equilibrium of different systems. There has been some proposal to consider it as a consequence of the first and second law in the past. Even recently, it has been argued that thermodynamics and the concept of temperature could be developed without the zeroth law (Kammerlander, P., & Renner, R. (2018)).
However, there is something in the zeroth law that goes beyond the definition of temperature. Some decades ago, Redlich ( Journal of Chemical Education, 47(11), 740 (1970)) observed that if we found a case where three systems having the same temperature are not in mutual thermal equilibrium, we would not discard the concept of temperature. Still, we would look for unknown interactions that have not been screened to leave only a thermal contact. In this respect, the zeroth principle still plays a role as a guideline about how we perform experiments.
