  We study the issue of temperature in a steady system around a black hole
event horizon, contrasting it with the appearance of divergence in a thermal
equilibrium system. We focus on a spherically symmetric system governed by
general relativity, particularly examining the steady state with radial heat
conduction. Employing an appropriate approximation, we derive exact solutions
that illuminate the behaviors of number density, local temperature, and heat in
the proximity of a black hole. We demonstrate that a carefully regulated heat
inflow can maintain finite local temperatures at the black hole event horizon,
even without considering the back-reaction of matter. This discovery challenges
conventional expectations that the local temperature near the event horizon
diverges in scenarios of thermal equilibrium. This implications shows that
there's an intricate connection between heat and gravity in the realm of black
hole thermodynamics.
