You probably haven't been facing the problems associated with amplitude varying with BW, then.
The OP wants to build an All Pass filter. The common topology uses a filter (could be Lp, Hp or Bp) that has unity gain, which is substracted to the input signal in a 2:1 ratio. The result is a filter that has flat amplitude all over, but has variable phase. The consequence of variable phase is variable propagation time. AP filters are often used in a range where the time delay is pretty much constant. Until Bucket Brigade Delays and DSP became practical, AP filters were the only solution to produce delay electronically (tape/disk is electromechanical and acoustic delay lines are...electro-acoustic).
The amount of delay is usually very small (about 5us for a 20kHz BW).
AP filters were used, among other applications, for compensating the 22us delay between the left and right outputs of early stereo DAC's.
They are still used today for alignment of speaker systems, in particular to compensate the phase-shift introduced by cross-overs in multiway systems.