  In this paper, we investigate the angular momentum independence of the
entropy sum and product for AdS rotating black holes based on the first law of
thermodynamics and a mathematical lemma related to Vandermonde determinant. The
advantage of this method is that the explicit forms of the spacetime metric,
black hole mass and charge are not needed but the Hawking temperature and
entropy formula on the horizons are necessary for static black holes, while our
calculations require the expressions of metric and angular velocity formula. We
find that the entropy sum is always independent of angular momentum for all
dimensions and the angular momentum-independence of entropy product only holds
for the dimensions $d>4$ with at least one rotation parameter $a_i=0$, while
the mass-free of entropy sum and entropy product for rotating black holes only
stand for higher dimensions ($d>4$) and for all dimensions, respectively. On
the other hand, we find that the introduction of a negative cosmological
constant does not affect the angular momentum-free of entropy sum and product
but the criterion for angular momentum-independence of entropy product will be
affected.
