Violation of the First Law of Thermodynamics in f(R, T) Gravity

We derive the first law of thermodynamics using the method proposed by Wald. Treating the entropy as Noether charge and comparing with the usual first law of thermodynamics, we obtain explicitly the expression of entropy which contains infinitely many non-local terms (i.e. the integral terms). We have proved, in general, that the first law of black hole thermodynamics is violated for f(R, T) gravity. However, there might exist some special cases in which the first law for f(R, T) gravity is recovered.