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In this paper, we investigate the angular momentum independence of theentropy sum and product for AdS rotating black holes based on the first law ofthermodynamics and a mathematical lemma related to Vandermonde determinant. Theadvantage of this method is that the explicit forms of the spacetime metric,black hole mass and charge are not needed but the Hawking temperature andentropy formula on the horizons are necessary for static black holes, while ourcalculations require the expressions of metric and angular velocity formula. Wefind that the entropy sum is always independent of angular momentum for alldimensions and the angular momentum-independence of entropy product only holdsfor the dimensions $d>4$ with at least one rotation parameter $a_i=0$, whilethe mass-free of entropy sum and entropy product for rotating black holes onlystand for higher dimensions ($d>4$) and for all dimensions, respectively. Onthe other hand, we find that the introduction of a negative cosmologicalconstant does not affect the angular momentum-free of entropy sum and productbut the criterion for angular momentum-independence of entropy product will beaffected.

Angular Momentum Independence of the Entropy Sum and Entropy Product for AdS Rotating Black Holes In All Dimensions