It has been shown that the fundamental thermodynamic relation together with the following three postulates
is sufficient to build the theory of statistical mechanics without the equal a priori probability postulate.
For example, in order to derive the Boltzmann distribution, we assume the probability density of microstate  satisfies . The normalization factor (partition function) is therefore
The entropy is therefore given by
If we change the temperature  by  while keeping the volume of the system constant, the change of entropy satisfies
where
Considering that
we have
From the fundamental thermodynamic relation, we have
Since we kept  constant when perturbing , we have . Combining the equations above, we have
Physics laws should be universal, i.e., the above equation must hold for arbitrary systems, and the only way for this to happen is
That is
It has been shown that the third postulate in the above formalism can be replaced by the following:
However, the mathematical derivation will be much more complicated.