Supposing the existence of dark fluid with a Chaplygin-like equation of state $p=-B/\rho$ (CDF) as a cosmic background, we obtain a static spherically symmetric black hole (BH) solution to the Einstein gravitational equations. We study the $P-V$ critical behavior of an anti-de Sitter (AdS) BH surrounded by the CDF in the extended phase space where the cosmological constant appears as pressure, and our results show the existence of the van der Waals like small/large BH phase transition. Also, it is found that such a BH displays a first-order low-/high-$\mathrm{\Phi }$ BH phase transition and admits the same criticality with the van der Waals liquid/gas system in the nonextended phase space, where the normalization factor $q$ is considered as a thermodynamic variable, while the cosmological constant is fixed. In both $P-V$ and the newly proposed $q-\mathrm{\Phi }$ phase spaces, we calculate the BH equations of state and then numerically study the corresponding critical quantities. Moreover, the critical exponents are derived, and the results show the universal class of the scaling behavior of thermodynamic quantities near criticality. Finally, we study the shadow thermodynamics of AdS BHs surrounded by the CDF. We find that there exists a positive correlation between the shadow radius and the event horizon radius in our case. By analyzing the temperature and heat capacity curves under the shadow context, we discover that the shadow radius can replace the event horizon radius to demonstrate the BH phase transition process, and the changes of the shadow radius can serve as order parameters for the small/large BH phase transition, indicating that the shadow radius could give us a glimpse into the BH phase structure from the observational point of view.

• Received 17 October 2022
• Accepted 3 May 2023

DOI:https://doi.org/10.1103/PhysRevD.107.104055