Fabry-Perot (FP) cavities having at least one partially reflective surface are widely used as compact and efficient high-directivity radiators. However, they inherently suffer from very narrow directivity bandwidths around the FP resonance. Here it is shown that this constraint can be overcome by using an active non-Foster cavity boundary behaving as a series negative LC tank circuit. Analytical and full-wave numerical results reveal the possibility of nearly tenfold improvement of - 1 dB directivity bandwidth while preserving the original overall antenna size and directivity of the antenna.