Barrow Entropy and the Reshaped Phase Structure of a Charged Anti–de Sitter Black Hole in Massive Gravity

The Bekenstein–Hawking area law is the one firm prediction that every approach to quantum gravity must reproduce, and any controlled departure from it is a probe of the horizon’s microscopic constitution. We take the departure proposed by Barrow, in which quantum–gravitational fluctuations endow the horizon with a fractal, area–enhancing roughness so that the entropy becomes the power law [Formula: see text], and we work out its consequences for the four–dimensional charged anti–de Sitter black hole of de Rham–Gabadadze–Tolley massive gravity in the extended phase space. Because the Barrow term grows with horizon size rather than saturating, it acts across the whole thermodynamic range and most strongly at large and intermediate horizons, in sharp contrast to perturbative logarithmic or bounded exponential corrections whose weight lies at the smallest scales. Adopting the entropy–conjugate temperature [Formula: see text], which alone keeps the enthalpy identification and the extended first law intact, we show that the fractal exponent enters the equation of state directly and reorganizes the entire phase diagram. From a single set of independently verified closed forms we obtain the Smarr relation, the corrected potentials, the heat capacity, the [Formula: see text]–[Formula: see text] criticality, the coexistence and spinodal curves, the latent heat, the critical exponents, the Joule–Thomson inversion, the Ruppeiner curvature and a benchmark heat engine. The critical point migrates strongly with [Formula: see text], the coexistence line lengthens, the compressibility ratio [Formula: see text] abandons its van der Waals value, and the Joule–Thomson inversion condition is displaced from the classical factor three to [Formula: see text]; the critical exponents remain mean–field but with [Formula: see text]–dependent amplitudes, and the Ruppeiner spinodal moves to larger volumes with a deepened attractive well. Against these shifts one observable stands invariant: in the thermodynamically consistent scheme the engine efficiency is exactly independent of [Formula: see text], a fact we trace to the geometric character of the enthalpy. The area law and the massless, neutral and topological limits are recovered analytically. A fractal horizon, we conclude, does not decorate the classical phase diagram but redraws it, with [Formula: see text] a genuine control parameter over the black hole’s thermodynamic response.