Klein-Gordon equation is the starting point of many calculations for black holes. One can write the radial part of the Klein-Gordon equation in a specific form that reveals the pole structure of the horizon equation. The residua of the poles reflect the physical
properties, namely surface gravities and angular velocities associated with respective horizons. Generally one has four or more singularities for the radial equation but in some limiting cases it reduces to a hypergeometric equation. This reduction signifies the conformal invariance.
I will start with a very brief introduction to black hole thermodynamics and Hawking radiation. Then I will focus on the most general non-extremal black hole solutions in four and five dimensional gauged supergravity in the literature to show the specific structure of the radial part of the Klein-Gordon equation. I will finish with comments on the emergence of conformal invariance under geometrical and non-geometrical limits.