  We investigate the critical behavior of an $(n+1)$-dimensional topological
dilaton black holes, in an extended phase space in both canonical and
grand-canonical ensembles, when the gauge field is in the form of power-Maxwell
field. In order to do this we introduce for the first time the counterterms
that remove the divergences of the action in dilaton gravity for the solutions
with curved boundary. Using the counterterm method, we calculate the conserved
quantities and the action and therefore Gibbs free energy in both the canonical
and grand-canonical ensembles. We treat the cosmological constant as a
thermodynamic pressure, and its conjugate quantity as a thermodynamic volume.
In the presence of power-Maxwell field, we find an analogy between the
topological dilaton black holes with van der Walls liquid-gas system in all
dimensions provided the dilaton coupling constant $\alpha$ and the power
parameter $p$ are chosen properly. Interestingly enough, we observe that the
power-Maxwell dilaton black holes admit the phase transition in both canonical
and grand-canonical ensembles. This is in contrast to RN-AdS,
Einstein-Maxwell-dilaton and Born-Infeld-dilaton black holes, which only admit
the phase transition in the canonical ensemble. Besides, we calculate the
critical quantities and show that they depend on $\alpha$, $n$ and $p$.
Finally, we obtain the critical exponents in two ensembles and show that they
are independent of the model parameters and have the same values as mean field
theory.
