Stefan's law tells gives an expression for thermal radiation emitted per unit time by a body of surface area A and temperature $T$
$$ u = \sigma A e T^4$$
In my book, it is written that in thermal equilibrium the energy of a body radiated out by stefan's law is equal to the energy radiated out. So, if a body is initially at temperature $T_o$ then it's absorbed heat is given by :
$$ u = \sigma A e (T_o)^4$$
And now suppose the body has it's temperature raised but room temperature is constant, then the energy radiated out per unit time is $ u = e \sigma A T^4$ and the absorbed is said to be $ u_o = e \sigma (T_o)^4$
Now, this is where I'm confused, why is the heat absorbed out at thermal equilibrium the same amount which is absorbed out when the temperature of the body is raised? Is there a proof for this?
Reference: In Concepts of Physics part-2 by H.C.-Verma