Why is $\int_{0}^{2\pi} \int_0^{2\pi} \frac{\ln(21-4(\cos x+\cos y+\cos(x+y)))}{2\ln(9/2)}\frac{dx}{2\pi} \frac{dy}{2\pi}$ almost $1$?

Question

Consider the function $$ f(x,y) = \frac{\ln(21-4(\cos(x)+\cos(y)+\cos(x+y)))}{2\ln(9/2)} $$

Its average value is awfully close to unity: $$ \int_{0}^{2\pi} \int_0^{2\pi} f(x,y) \frac{\mathrm dx}{2\pi} \frac{\mathrm d y}{2\pi} = 1.00095 $$ (This result is rounded.)

Is this just a fluke, or is there some deeper reason to it?