Ratio of critical quantities related to Hawking temperature-entanglement entropy criticality

We revisit the Hawking temperature$-$entanglement entropy criticality of the $d$-dimensional charged AdS black hole with our attention concentrated on the ratio $\frac{T_c \delta S_c}{Q_c}$. Comparing the results of this paper with those of the ratio $\frac{T_c S_c}{Q_c}$, one can find both the similarities and differences. These two ratios are independent of the characteristic length scale $l$ and dependent on the dimension $d$. These similarities further enhance the relation between the entanglement entropy and the Bekenstein-Hawking entropy. However, the ratio $\frac{T_c \delta S_c}{Q_c}$ also relies on the size of the spherical entangling region. Moreover, these two ratios take different values even under the same choices of parameters. The differences between these two ratios can be attributed to the peculiar property of the entanglement entropy since the research in this paper is far from the regime where the behavior of the entanglement entropy is dominated by the thermal entropy.