Isn't energy absolute according to Thermodynamics? I was taught that Internal Energy $U$ is  a relative quantity: only changes in $U$ are meaningful. It doesn't have an absolute value, since it always comes with an arbitrary constant (for example $U = nT +c$).
Entropy $S$, on the other hand, has an absolute value thanks to the Third Postulate of Thermodynamics. Volume $V$ and number of moles $n$ are also absolute, obviously.
The Fundamental Relation of a system relates $S$, $U$, $V$ and $n$ (For example $U = e^{S/n}Vn$, but any example would work).
Since $S$, $V$, and $n$ are always absolute quantities, doesn't that mean that $U$ must also be absolute? So any system has a meaningful, absolute value of $U$ with no arbitrary constant?
 A: It's true that the temperature $T$, entropy $S$, pressure $P$, volume $V$ and amount of material $N$ are all amenable to being modeled as absolute. However, thermodynamics formulates the complete expression for internal energy as
$$U=TS - PV + \sum \mu N.$$
This shifts the relative nature to the chemical potential $\mu$, which—like the internal energy $U$—can be measured relative to a reference only. (The same holds for the enthalpy $H\equiv U+PV$, the Helmholtz potential $F\equiv U-TS$, and the Gibbs free energy $G\equiv U+PV-TS$, and after all, the chemical potential $\mu\equiv\left(\frac{\partial G}{\partial N}\right)_{T,P}$ is just the molar Gibbs free energy.) If you consider a closed system at equilibrium so that the last term is constant, then you’re implicitly taking that system as the reference.
A: Energy and entropy are not directly measurable. Only differences may be measured. It is enough to think about typical energy measurements via work and heat measurements (always referring to an initial and a final state).
The third principle of thermodynamics only states that the limit of the entropy for temperature going to absolute zero is a constant independent of the material and other thermodynamic variables. Then, due to the arbitrariness of the constant (only differences are measurable), it is convenient to take it equal to zero (this final step was taken by Planck, who explicitly wrote, "It is easy to see that since the value of the entropy contains an arbitrary additive constant, that, without loss of generality,
we may write $\lim_{T\rightarrow 0}S(T) = 0$.  M.Planck, Treatise on Thermodynamics).
In essence, the real content of the third principle is that entropy is bounded below. This fact strictly parallels the similar property for the energy of any thermodynamical system. The exact value of such a lower limit can be conventionally fixed to any value since has no measurable consequence. The most convenient choice is zero.
I have to say that the convenience of bounding from below energy and entropy with zero is usually underestimated in the classical presentations of thermodynamics. Actually, it simplifies many statements that would become unduly complicated if the range of values of the entropy $S$ and internal energy $U$ would not start from zero. It is enough to think how simple it is to state the condition of homogeneity of degree one (extensiveness) of $S$ (or $U$) as a function of its extensive variables:
$$
S(\lambda U, \lambda V, \lambda N) = \lambda S(U,V,N)
$$
for every $U$, $V$, $N$, and $\lambda$ greater than zero.
Here, the domain of $S(U,V,N)$ can be taken as the cone $U>0, V>0$, and $N>0$. If we would not take the minimum of $U$ as zero, a value $\lambda>1$ could bring a value of energy close to the minimum out of the original domain of $S$. Of course, we could restate the condition more carefully, but at the price of a more complex statement.
Statistical mechanics could induce us to think that the value of the entropy is absolute. However, it is again only a matter of convention. It is pretty clear in the case of classical statistical mechanics, where the counting of the microstates is possible only by introducing a constant $h$, with the dimension of an action, whose value is arbitrary since it modifies the value of the entropy only by a universal constant. Quantum Statistical Mechanics seems to establish an absolute value of the entropy. However, in this case, too, one quickly recognizes that the identification of the statistical formula for the logarithm of the number of microstates with the physical entropy requires integrating the relation
$$
\frac{1}{k_BT} = \frac{\partial \log \Omega}{\partial E}
$$
resulting once again in an arbitrary constant.
