Why is the zeroth law necessary when temperature is simply a derived quantity in the microcanonical ensemble? The Zero'th Law of Thermodynamics states that : If two systems are in thermal equilibrium with a third system, then they are in thermal equilibrium with each other.
Clearly, this introduces an equivalence relation between all thermodynamic systems and as a result, it is frequently stated that the purpose of the zeroth law is to recognize this empirical fact and introduce the notion of temperature $T$ to characterize the equivalence classes. My first issue is that if a zeroth law introducing temperature ($T$) as characterizing an equivalence class is necessary for the logic of thermodynamics, then why do we not have 4th and 5th law's introducing chemical potential and pressure as characterizing equivalence classes? For example, we could right the 4th law as : If two systems are in diffusive equilibrium with a third system, then they are in diffusive equilibrium with each other.
Why is the zero'th law and the notion of temperature needed for the internal logic of thermodynamics to make sense when my supposed 4th law is not needed? What makes temperature so difference from pressure or chemical potential in this regard?
Secondly, from my understanding, fundamental laws in physics should meet two criteria in order to be regarded as fundamental laws. Firstly, they should be statements that comport with empirical evidence to such an extent that we regard them as accepted truths. And secondly, they should not be derivable from any other statements or fundamental laws. If they are derivable from some other statement, then they can be discarded and the statement from which they are derived can be classified as fundamental instead.
If we look through the perspective of the microcanonical ensemble though, temperature is simply a derived quantity defined by $\frac{1}{T}\equiv\left(\frac{\partial S}{\partial U}\right)_{V,N}$. In this ensemble, we absolutely need the first and the second law's of thermodynamics. That is, we need to assume energy conservation, and we need to assume that the entropy of an isolated system never decreases. But once we've made these two assumptions, we get the notion of temperature for free. Similarly, if we allow 2 subsystems of an isolated system to alter their volume, from the first and second laws alone, we get the notion of pressure for free $\left(\frac{P}{T}\equiv\left(\frac{\partial S}{\partial V}\right)_{U,N}\right)$. The same goes for diffusion and chemical potential. It seems to me that the introduction of temperature as a state variable through the zeroth is not on equal footing with the introduction of energy as a state variable (via the first law) or entropy as a state variable (via the second). Its importance is merely equivalent to that of pressure or chemical potential? What about this am I missing? Why is temperature and the zeroth law so crucial to thermodynamics even though it seems to be derivable from the 1st and 2nd laws whilst pressure and chemical potential are not deemed to be so crucial?
 A: There is a school of thought that attempts to formulate classical thermodynamics in terms of a minimum set of axioms so as to make all other results of thermodynamics inevitable. This approach strives for logical consistency though not for pedagogy, let alone for physical insight. And when it comes to "axioms", one is left to decide what constitutes an axiom versus what constitutes knowledge that exists prior to the axiom. Since the axiomatic approach is divorced from physics, it needs to express in axioms what the physicist considers "obvious" based in empirical observation.
One could say that the "zeroth" law, which was added to the list of axioms long after it had been routinely and unquestionably accepted as obvious, to express the fact that if a system is in equilibrium, then every one of its internal parts is in equilibrium with all others. Thus we can speak of a uniform temperature, pressure and chemical potential across the entire system.
At the end of the day one does not need axioms to describe nature. One needs observations and a quantitative hypothesis (theory) that ties the observations together. Statistical mechanics does that beautifully.
Statistical mechanics is built not on axioms, but on a physical hypothesis: that all microstates with the same energy, volume and number of particles contribute equally to the mechanical properties of a system in equilibrium. This is and remains a physical hypothesis because there is no way to measure the probability of microstates experimentally. The validity of stat mech is contingent upon evidence: as long as its predictions agree with measurement, we accept the theory as valid.
A: I think you inverted the logic a bit when showing how $T$ can be derived from entropy. In fact it is the opposite. In classical thermodynamics you define entropy as $d S=\delta Q/T$.
You may argue that entropy can be defined statistically (with no regards for thermal state) when you go down into microscopics but in fact here it is even more complicated and confusing. The existing dynamical theories (classical or quantum mechanics) are reversible. They do not accept probabilities. In the absence of a ${\it thermal}$ bath the system is always in a pure state, and hence its entropy is zero. Not a very useful quantity.
But when there is a bath there is an entropy one may say. True, but what is a bath if not another system in a thermal equilibrium? How did ${\it that}$ system enter a ${\it thermal}$ state and what does it even mean?
So to me, the statement of the zeroth Law of Thermodynamics (since it is not a canonic law different people formulate it differently, and I admit my understanding might not match what you have in your mind) is that ${\it there\,  exists}$ a thermal state with the properties that you have described.
How obvious it is? Well, I believe it is not. For instance, there is a very active sub-field in condensed matter physics that works what they call "many body localization". What this means is that under certain circumstances the interactions in a many-body system "break ergodicity" and thus prohibit it from reaching a thermal state.
P.S. As for pressure and other quantities such as chemical potential, to me, they are not on equal footing with temperature, as all of them have a very transparent meaning that can be derived from pure mechanics. Whereas temperature is different. As I tried to articulate it above, it (or the very idea of a thermal state) doesn't really fit the rest of physics, so its existence has to be postulated.
