Black Hole Solutions in Three-Dimensional Gravity within the Framework of the Modified F(Q, T) Gravity Model

We investigate static black hole solutions in three-dimensional ( 2 + 1 ) spacetime within the framework of modified F ( Q, T ) gravity, where the gravitational action depends on the non-metricity scalar Q and the trace of the energy-momentum tensor T . Considering a linear model F ( Q , T ) = C 1 Q + C 2 T , we derive the modified field equations under a Schwarzschild-like gauge and obtain both perturbative and numerical solutions. The weak-field expansion is shown to be valid in the asymptotic regime | a ( r )| ≪ 1, while the near-horizon structure is analyzed through full numerical integration of the coupled differential system. The horizon is defined consistently in 2 + 1 dimensions via A ( r + ) = 0 , and the Hawking temperature is obtained from surface gravity without invoking four-dimensional Schwarzschild relations. Using the Wald formalism adapted to non-metricity gravity, we derive the entropy and show that it reduces to S = π C 1 r + / 2 , recovering the BTZ result in the limit C 1 = 1 . A general expression for the heat capacity is obtained, demonstrating that its sign depends on the second radial derivative of the metric function. While BTZ-like geometries yield positive heat capacity and thermodynamic stability, the numerical F ( Q, T ) solutions exhibit regions where A ″ ( r + ) < 0 , leading to negative heat capacity and instability. Our results show that although the leading-order horizon structure coincides with BTZ geometry, the matter non-metricity coupling significantly modifies the radial structure and thermodynamic stability. This demonstrates that F ( Q, T ) gravity in three dimensions admits black hole solutions that are geometrically BTZ-like but thermodynamically distinct.