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

Counterterm method in dilaton gravity and the critical behavior of dilaton black holes with power-Maxwell field