Total emitted power of diffuse area light
Radiance (in terms of flux) has the following definition:

$L_o = \frac{\mathrm{d}\Phi}{\mathrm{d}\omega^\perp\mathrm{d}A}
= \frac{\mathrm{d}\Phi}{\cos{\theta} \mathrm{d}\omega \mathrm{d}A}$.

Thus in order to get the total emitted power we need to integrate over the area of the light and we need to integrate over the projected solid angle of the hemisphere (in the direction of the surface normal). The radiance for the area lights in pbrt is constant. This gives us:

$\Phi = \int_A \int_{H^2(\mathbf{n})} L_o\mathrm{d}\omega^\perp \mathrm{d}A =
\int_A \int_{H^2(\mathbf{n})} L_o \cos\theta \mathrm{d}\omega \mathrm{d}A =
\int_A L_o \int_0^{2\pi} \int_0^{\pi/2} \sin\theta \cos\theta \mathrm{d}\theta \mathrm{d}\phi \mathrm{d}A =
\int_A L_o \pi \mathrm{d}A =
A \cdot L_o \cdot \pi$