  For black holes with multiple horizons, the area product of all horizons has
been proven to be mass independent in many cases. Counterexamples were also
found in some occasions. In this paper, we first prove a theorem derived from
the first law of black hole thermodynamics and a mathematical lemma related to
the Vandermonde determinant. With these arguments, we develop some general
criterion for the mass independence of the entropy product as well as the
entropy sum. In particular, if a $d$-dimensional spacetime is spherically
symmetric and its radial metric function $f(r)$ is a Laurent series in $r$ with
the lowest power $-m$ and the highest power $n$, we find the criteria is
extremely simple: The entropy product is mass independent if and only if $m\geq
d-2$ and $n\geq4-d$. The entropy sum is mass independent if and only if $m\geq
d-2$ and $n\geq 2$. Compared to previous works, our method does not require an
exact expression of the metric.
  Our arguments turn out to be useful even for rotating black holes. By
applying our theorem and lemma to a Myers-Perry black hole with spacetime
dimension $d$, we show that the entropy product/sum is mass independent for all
$d>4$, while it is mass dependent only for $d=4$, i.e., the Kerr solution.
