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

The relationship between thermodynamics and the Lloyd bound on the holographic complexity for a black hole has been of interest. We consider D dimensional anti--de Sitter black holes with hyperbolic geometry as well as black holes with momentum relaxation that have minimum values for the temperature and the mass. We will show that the singular points of the thermodynamic curvature of the black holes, as thermodynamic systems, correspond to the zero points of the action and volume complexity at the Lloyd bound. For such black holes with a single horizon, the rates of growth of the complexity of volume and the complexity of action at minimum mass and minimum temperature are zero, respectively. We show that the thermodynamic curvature diverges at these minimal values. Because of the behavior of the growth rate of the action complexity and thermodynamic curvature at minimum temperature, we propose the growth rate of the action complexity as an order parameter of the black holes as the thermodynamic systems. Also, we derive the critical exponent related to the thermodynamic curvature in different dimensions.