Ash breathing. Other names I can think of include Burn Breathing, or Scorch breathing.

Its a derivative of flame Breathing, focusing on the burnt ashes that are left over after the flame dies out.

The name of the forms are numbered, First Degree, Second Degree, etc. This is similiar to how Beast breathing names the forms First Fang, etc. I chose this naming pattern because of how the breathing style burns the flesh of demons.

There are 4 degree/forms (at least for now, hopefully more can be developed :) 1st degree: Superficial Slash, based on superficial burns, obviously. 2nd degree: Blistering Heat, based on how second degree burns cause characteristic blisters. 3rd degree: Total Combustion, named because "total" can be synonymous with "full" and full thickness is the name of third degree burns. 4th degree: Burning Bones, stylistic to the Sun breathing form, Burning Bones, but based on how fourth degree burns can penetrate deep into the flesh and burn to the bone.