Is the antenna gain time dependent?
If you had an amplifier with a gain of 5 and fed to its input a sinewave of 1 volt RMS, you would expect to see on the output a 5 volts RMS sinewave. The fact that a 1 volt RMS sinewave has instantaneous values ranging from -1.4142 volts to +1.4142 does not alter the fact that the amplifier gain remains constant and produces an output sinewave that ranges between -7.071 volts and +7.071 volts.

Is it a time varying quantity?

No.