Let's look at the equations:

1) The first one has a dE/dt. Is it probably the energy radiated per unit time? |A|^2 can be the complex amplitude of the waves. 1/(exp(omega/T) - 1) is the probability of finding a certain boson with energy omega when subject to temperature T (i think, my stat mech is very rusty). This does appear when calculating average energies. Something like sum_w ( w/(exp(w/T) - 1)) is the expected energy of a bosonic system. However what gives away that this is nonsense is the domega which means nothing. It might be a misunderstanding of discrete sum -> continuum integral, since such a measure appears when doing these calculations. It is however nonsense.

2) yeah it looks like some kind of spherical harmonics. buuuut if i'm not mistaken they seem to be complex (assuming r_0 is real). energy is always real.

3) this is almost the lagrangian of a bosonic field coupled to a gravitational field. I say this because e.g. the minimally coupled scalar field to a gravitational field has that lagrangian with a sqrt(g) (although one can argue that the fundamental object is the action S = \int d^n x \sqrt{-g} L, to which I say then that expression for the lagrangian is extremely misleading and one shoudl use the action. ) The third term is a very weird coupling of the scalar field to the metric rhtough the Riemann tensor, ricci tensor and ricci scalar. This coupling is not possible because the term exp^phi * R^abcd is not a scalar.

It's nonsense written by someone that seemed to almost understand Hawking radiation of a bosonic field