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Strange question. Let's try a thought experiment. Let's say you have a resistor(R) from input to output, and your output is infinite impedance. Infinite bandwidth, and current consumption is zero. Now let's say you put a cap across the load (you now have a low pass filter). At very high frequencies your current is Vin/R; at very low frequencies current =zero (again). So your current is a function of your INPUT frequency, not your bandwidth. I think you need to analyze your particular circuit. And what do you mean by 'empirical relationship'? By definition, empirical means by determined solely by experimentation or observation, NOT theory.
There is a quick equation for predicting your current consumption with GBW. If your GBW= gm/(2*pi*Cc), where Gm is the gain in your first stage input transistors, and Cc is your compensation cap. A good starting point for amp design is just making your Cc= 1pF. Ofcourse this can be changed depending on the load... Its just a rule of thumb. Then you have your gm = 2* Id/Vov, where Id is the current in your transistor and Vov is the overdrive in the transistor. So if you have an overdrive of say 200mV you have gm = 10*Id...
which leads to GBW = 10*Id/(2*Pi*1pF).
Of course this all changes on how you design your amp.. but this will put you in the "ballpark" of what to expect...
So with that you can have a current consumption to bandwidth relationship....
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