  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 a minimum for temperature and
mass. We 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 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
behaviour of action complexity and thermodynamic curvature at minimum
temperature, we propose the action complexity as an order parameter of the
black holes as thermodynamic systems. Also, we derive the critical exponent
related to the thermodynamic curvature in different dimensions.

High Energy Physics - Theory;General Relativity and Quantum Cosmology