Can entropy be intensive? Entropy typically is an extensive thermodynamic variable. Thus, if I combine two subsystems 1 and 2, the total entropy $S_{total} = S_1 + S_2$. This follows directly from the Boltzmann-entropy when we assume that the two subsystems are independent. In that case the partition sum of the full system is the product of the partition sum of the two subsystems:
$$
\Omega_{total} = \Omega_1 \cdot \Omega_2
$$
And thus
$$
S_{total} = k_B\ln\Omega_{total} = k_B\ln\Omega_1 + k_B\ln\Omega_2 = S_1 + S_2
$$
However, the assumption of independent subsystems might be not hold for all systems. For example, a phase transition is typically associated with a diverging correlation length, making independent subsystems impossible. The extreme case would be the perfect crystal, where the crystal orientation in one subsystem defines the orientation in any other subsystem. However, the perfect crystal has zero entropy, thus making it a bad example.
Do systems exist (as theoretical or artificial as they might be) for which the entropy is an intensive variable?
 A: The thermodynamics of small systems has entropy being intensive
http://aip.scitation.org/doi/pdf/10.1063/1.1732447
A: I didn't study non-equilibrium thermodynamics so some of my views might be too simplistic.
The classical definition by Clausius explicitly states that entropy should be an extensive quantity. Also entropy is only defined in equilibrium state.
If you take one container with oxygen and one with hydrogen their total entropy will be the sum of the entropies. If you mix the containers together the total entropy will increase because this is a non-equilibrium process. The entropy can further increase if the mixture ignites... After the system reaches equilibrium entropy is extensive again.
The total entropy grows in non-equilibrium processes. Equilibrium process is a limit of slow quasistatc processes. If the limit doesn't exist then the process will always be non-equilibrium.
It seems that entropy must be always extensive. The only way how it can be intensive is that it is intensive and extensive at the same time. Intensive is not oposite of extensive. There are quantities that are neither extensive nor intensive (but as far as I know none of them is a state variable). Intensive means equal in all subsystems. Hence following equation must be satisfied for the system where entropy is both extensive and intensive:
$$S_{tot}=S_1+S_2=S_1=S_2$$
The equation can only be satisfied if all the quantities are zero.
Entropy is intensive in systems with zero entropy. I dare say that only in those systems.
