In this work, we obtain the black hole solutions in the dilaton f(R)-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 f(R)-gravity which is the basis of our work. We also obtain the exact form of f(R) 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.

中文翻译：

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膨胀子 f(R)-引力中的 BTZ 黑洞解会发生什么？

在这项工作中，我们获得了膨胀子中的黑洞解F(R)-重力（R在这里不被视为常数）并研究它们的热力学，特别是反德西特（AdS）扩展相空间中的相变和临界行为。我们在扩张子中获得了精确的 Banados、Teitelboim 和 Zanelli (BTZ) 对应解F(R)-重力，这是我们工作的基础。我们还得到了的确切形式F(R)一些解决方案的模型。在热力学分析中，我们计算这些解的温度和熵等热力学量，并将它们与 BTZ 对应量进行比较。之后，我们研究这些获得的解决方案的稳定性（局部和全局）。在批判行为分析中，我们发现没有证据表明存在磷–五在这个修改后的重力模型中的临界（像普通的 BTZ 案例），除了一些不寻常的磷–五相应图表中的行为。