What is the physical interpretation of the absolute value of a fourier transformed signal, $\left| F(t)\right|$?
This would be the so called magnitude spectrum, its unit is the same as the signal it's generated from, in the case of voltage its interpretation is "voltage  by time over frequency" as opposed to "power by time over frequency" of the squared magnitude spectrum. 
In the ususal logarithmic representation, the two are proportional by a factor of 2 and essentially give you the same information.

The $\cos$ function in your example is a so called power signal, not an energy signal, as it is infinitly long it has infinite energy, so to assess its energy in a practical way, one refers to its power (energy per time).