In this work, we obtain the black hole solutions in the dilaton [Formula: see text]-gravity (R is not considered as a constant here) and investigate their thermodynamics especially phase transition and critical behavior in the anti-de Sitter (AdS) extended phase-space. We obtain the exact Banados, Teitelboim and Zanelli (BTZ) counterpart solutions in dilaton [Formula: see text]-gravity which is the basis of our work. We also obtain the exact form of [Formula: see text] model for some solutions. In the thermodynamical analysis, we calculate the thermodynamical quantities like the temperature and entropy for these solutions and we compare them with the BTZ corresponding quantities. After that, we investigate the stability (local and global) for these obtained solutions. In the critical behavior analysis, we find that there is no evidence to show the existence of P–V criticality (like the ordinary BTZ case) in this modified gravity model except some unusual P–V behavior in the corresponding diagrams.

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