Recall that a black hole is a Planck emitter (or blackbody). The amount of energy lost by a Planck emitter per unit area per unit time (the energy flux, F) is given by the Stefan-Boltzmann equation:
\begin{equation}F=\sigma T^4\end{equation}
where the constant σ = 5.67 × 10–8 W/m2K4 and is needed to account for the system of units we use.
The relationship between flux and luminosity is: F = L/(4\pi r^2). This means that the total amount of energy emitted by the black hole, its luminosity, L, is this energy flux multiplied by the total area of the event horizon, essentially the area of the black hole, ABH. Since a black hole is spherical, we can use the area of a sphere: A = 4πR2.
\begin{equation}L=A_{BH}\sigma T^4=4\pi R^2_{Sch}\sigma T^4\end{equation}
We can now substitute into this expression for the Schwarzschild radius, the appropriate radius for the event horizon of a black hole, and the temperature of the black hole to find the black hole thermal luminosity:
\begin{equation}
L_{th}=4\pi\sigma\left( \frac{2GM}{c^2}\right)^2 \left( \frac{h c^3}{16\pi^2 G M k_B}\right)^4
\end{equation}
Or simplifying:
\begin{equation} L_{th}=\frac{\sigma h^4c^8}{2^{12}\pi^7G^2k_B^4M^2} \end{equation}
This expression looks complicated, but it is mostly just a bunch of physical constants, or in other words, a product of a bunch of numbers. That means it will simplify to a single number divided by the square of the black hole mass. In SI units, it becomes:
\begin{equation} L_{th}=\frac{3.56\times10^{32}}{M^2} \end{equation}
where the thermal luminosity L is in W, the mass M is in kg, and the constant is in W kg2.
We have found the thermal luminosity of a black hole, the amount of energy lost to the black hole per unit time due to Hawking radiation. Now we would like to determine how long it can radiate its energy. That depends on how much energy it has to radiate. The total amount of energy in the black hole is contained in its rest mass: E = Mc2. We can estimate its lifetime, t, if we divide the energy in the black hole by its thermal luminosity. This is analogous to the way we estimate the lifetime of a star.
\begin{equation} t\approx\frac{E}{L} =\frac{Mc^2}{\left(\frac{\sigma h^4c^8}{2^{12} \pi^7G^2k_B^4M^2}\right)} \end{equation}
Simplifying, we have:
\begin{equation} t \approx\frac{2^{12}\pi^7G^2k_B^4M^3}{\sigma h^4 c^6} \end{equation}
Again this looks complicated, but we can simplify further by multiplying out all the constants as we did previously:
\begin{equation} t \approx 2.5\times 10^{-16}M^3 \end{equation}
where the lifetime t is in seconds, the mass M is in kg, and the constant is in s/kg3.