(Thermal) Luminosity and Temperature relation

In a given Eddington rate, the luminosity (in black body) should relate with the temperature of the origin.
This is a very robust way to understand the relationship.

First, the luminosity can be written as

(1)
\begin{align} L &= 4\pi R^{2}\,\sigma\,T^{4} \\ &= 4\pi a^{2}\, R_{s}^{2} \,\sigma\,T^{4} \end{align}

where $R \equiv a\,R_{s}$, and $R_{s}$ is Schwarzschild radius,

(2)
\begin{align} R_{s} = \frac{2\,G\,M_{BH}}{c^{2}}. \end{align}

So, the luminosity can be expressed as,

(3)
\begin{align} L = \frac{16\pi\,a^{2}\,G^{2}\sigma}{c^{4}}\,M_{BH}^{2}\,T^{4}. \end{align}

Since the eddington luminosity, $L_{Edd}$ is,

(4)
\begin{align} L_{Edd} = \frac{4\pi\,G M_{BH}\,m_{p}\,c}{\sigma_{T}}, \end{align}

one can eliminate the term of black hole mass in equation 3 by using equation 4.
From the equation 3, the black hole mass can be re-written as,

(5)
\begin{align} M_{BH} = \frac{\sigma_{T}}{4\pi\,G\,m_{p}\,c} L_{Edd} = \frac{\sigma_{T}}{4\pi\,G\,m_{p}\,c}\,\frac{L}{l}, \end{align}

where $l$ is eddington ratio $l \equiv L/L_{Edd}$.

Substituting the equation 5 into equation 3, then we can obtain the relationship between the luminosity and the temperature:

(6)
\begin{align} L=\frac{\pi\,m_{p}^{2}\,c^{6}\,l^{2}}{a^{2}\,\sigma\,\sigma_{T}^{2}} T^{-4} \end{align}

Therefore, $L \sim T^{-4}$

And, this result also can be interpret as
$M_{BH} \sim T^{-4}$ shown in (Mirabel 2002; Rees 1984), since $L \sim L_{Edd} \sim M_{BH}$.

Mirabel 2002
For a black hole accreting at the Eddington limit, the characteristic black body temperature at the last stable orbit in the surrounding accretion disk will be given approximately by T ∼ 2 × 10^7 M^−1/4 (Rees 1984), with T in K and the mass of the black hole, M, in solar masses. Then, while accretion disks in AGN have strong emission in the optical and ultraviolet with distinct broad emission lines, black hole and neutron star binaries usually are identified for the first time by their X-ray emission.