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 ($\beta = \frac{1}{\Omega} \frac{d\Omega}{dE}$) is equal for both systems.
 - 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.