The new black hole solution with anisotropic fluid in f ( R , L m , T ) gravity: Thermodynamics

In this work, we derive a new class of analytic black hole solutions within the framework of f ( R , L m , T ) gravity, where the black hole is surrounded by an anisotropic fluid acting as the matter source. We consider both linear and nonlinear forms of the function f ( R , L m , T ) , enabling a detailed exploration of how anisotropic pressures and different f ( R , L m , T ) influence the spacetime structure. Furthermore, we derive the conditions on the coupling parameters ( β 1 , β 2 , β 3 ) under which the energy condition is satisfied. Utilising these constraints, we then investigate the thermodynamic behaviour of the resulting black hole solutions in the presence of various matter fields. An important outcome of this study is that the results obtained deviate from those predicted by standard General Relativity. It is also observed that these deviations depend explicitly on the interaction between the matter Lagrangian L m and the trace T of the energy-momentum tensor.