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The relationship between thermodynamics and the Lloyd bound on theholographic complexity for a black hole has been of interest. We consider $D$dimensional anti-de Sitter black holes with hyperbolic geometry as well asblack holes with momentum relaxation that have a minimum for temperature andmass. We show that the singular points of the thermodynamic curvature of theblack holes, as thermodynamic systems, correspond to the zero points of theaction and volume complexity at the Lloyd bound. For such black holes with asingle horizon, the complexity of volume and the complexity of action atminimum mass and minimum temperature are zero, respectively. We show that thethermodynamic curvature diverges at these minimal values. Because of thebehaviour of action complexity and thermodynamic curvature at minimumtemperature, we propose the action complexity as an order parameter of theblack holes as thermodynamic systems. Also, we derive the critical exponentrelated to the thermodynamic curvature in different dimensions.

Thermodynamic geometry and complexity of black holes in theories with broken translational invariance