  Binary black hole (BBH) mergers, particularly those with component masses in
the pair-instability gap, may be produced by hierarchical mergers in the disks
surrounding Active Galactic Nuclei (AGN). While the interaction of an embedded
BBH with an AGN disk is typically assumed to facilitate a merger, recent
high-resolution hydrodynamical simulations challenge this assumption. However,
these simulations often have simplified treatments for the gas thermodynamics.
In this work, we model the possible consequence of various feedback from an
embedded BBH with a simple model that maintains an enhanced temperature profile
around each binary component. We show that when the minidisks around each BH
become hotter than the background by a factor of three, the BBH orbital
evolution switches from expansion to contraction. By analyzing the
gravitational torque profile, we find that this change in direction is driven
by a weakening of the minidisk spirals and their positive torque on the binary.
Our results highlight the important role of thermodynamics around BBHs and its
effect on their orbital evolution, suggesting that AGN disks could be efficient
factories for BBH mergers.
