  In this work we study the flow of holographic entanglement entropy in
dimensions $d \geq 3$ in the gauge/gravity duality set up. We observe that a
generalized entanglement temperature $T_g$ can be defined which gives the
Hawking temperature $T_H$ in the infrared region and leads to a generalized
thermodynamics like law $E= \left(\frac{d-1}{d}\right)T_g~S_{REE}$, which
becomes an exact relation in the entire region of the subsystem size $l$,
including both the infrared ($l\rightarrow\infty$) as well as the ultraviolet
($l\rightarrow 0$) regions. Furthermore, in the IR limit, $T_g$ produces the
Hawking temperature $T_H$ along with some correction terms which bears the
signature of short distance correlations along the entangling surface.
Moreover, for $d\geq 3$, the IR limit of the renormalized holographic
entanglement entropy gives the thermal entropy of the black hole as the leading
term, however, does not have a logarithmic correction to the leading term
unlike the BTZ black hole ($d=2$) case. The generalized entanglement
temperature $T_g$ also firmly captures the quantum mechanical to thermal
crossover in the dual field theory at a critical value $l_c$ of the subsystem
size in the boundary which we graphically represent for $AdS_{3+1}$ and
$AdS_{4+1}$ black holes. We observe that this critical value $l_c$ where the
crossover takes place decreases with increase in the dimension of the
spacetime.
