It depends on what temperature is. A simple statistical argument could go like this.
- the multiplicity of two systems is the product of their multiplicities
- in thermodynamic equilibrium the multiplicity is at its maximum
- $\frac{d}{dE}(\Omega_A \Omega_B) = 0$ for an infinitesimal transfer of internal energy from A to B
- the fractional change of multiplicity with internal energies ($\beta = \frac{1}{\Omega} \frac{d\Omega}{dE}$) must be equal for both systems (if $\Omega_A$ goes down by 1 ‰, $\Omega_B$ goes up by 1 ‰).
- identify $\beta = 1/kT$
If this is the case for systems A and B and for B and C, the fractional change of multiplicity with energy ($\beta$) should also be the same for A and C. At room temperature, it is about 4 % per meV, for any system.