Thermodynamic Stability of the New Black Hole Solution in f(R,G,T) Gravity

In this work, we construct a static, spherically symmetric black hole solution of f(R,G,T)=β1R+β2G2+β3T gravity surrounded by an anisotropic fluid, to first order in the Gauss–Bonnet coupling β2. Using the general two-function metric, we show that the β2G2 term necessarily induces a non-trivial lapse function, and we obtain the corrections to both metric functions in closed form. We compute the Wald entropy, Hawking temperature and heat capacity at fixed fluid parameters for the dust-like (w=0) and radiation-like (w=1/3) cases, and we verify the first law of black hole thermodynamics to first order in β2. The heat capacity is positive only within a bounded window of small-horizon radii, where the black holes are locally thermodynamically stable; larger black holes have negative heat capacity and are locally unstable.