Thermodynamic geometry of charged AdS black holes with a string cloud in Lorentz-violating Einstein–Kalb–Ramond gravity

Abstract We investigate the thermodynamic microstructure of electrically charged AdS black holes in Einstein–Kalb–Ramond bumblebee gravity in the presence of a spherically symmetric cloud of strings. Employing Weinhold and Ruppeiner thermodynamic geometries in complementary thermodynamic representations, we show that curvature singularities consistently track the spinodal boundaries and the second-order critical point associated with a van der Waals-like small/large-black hole phase transition. Moreover, the sign structure of the Ruppeiner curvature provides a transparent characterization of the competing microscopic interactions across the coexisting phases. We find that the Lorentz-violating parameter $$\ell$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ℓ</mml:mi> </mml:math> and the string-cloud parameter $$\alpha$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>α</mml:mi> </mml:math> shift the critical scales and rescale correlation measures while preserving the universality class of the critical behavior. We further comment on dynamical and holographic implications and contrast the thermodynamic sensitivity to $$(\ell ,\alpha )$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ℓ</mml:mi> <mml:mo>,</mml:mo> <mml:mi>α</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> with thin-disk optical signatures.