The AIM-54 was originally designed to be fitted to the F-111B but that project was cancelled. On a normal combat patrol the F-14 didn't carry the AIM-54 Phoenix because, at near $500,000 a missile, it was just too expensive. Maybe it's cost effective against the massive, expensive, long-range bombers of the Soviet Union - but they've gone.
The AIM-54 is also heavy. Fitting them to the Lightning would be extra weight that wouldn't be needed, after all the Lightning could intercept it's target far from the danger area. The weight would reduce the Lightning's capability rather than improve it.
The Phoenix, when launched at maximum range, was at risk of being jammed or confused from background clutter, even with the advanced AWG-9 fire-control. The BAe Sky Flash was a more advanced system that had less of a risk from clutter and could keep track on it's target more efficiently. The Raytheon AIM-7 Sparrow uses the same kind of system, and the U.S were interested in purchasing the Sky Flash instead of the Sparrow. Foreign users of the of the F-16 use the Sky Flash though.
While it only has twenty-eight mile range, the Sky Flash is more capable of striking it's target than the Phoenix which uses it's massive (and impressive) range to make a name for itself - range isn't everything. It still has to use the AWG-9 of the F-14 until the last ten miles too. So, the F-14 can't go home when it's fired all it's missiles off.
Did I forget to mention the F-14A had a ceiling of 56,000 feet? And the F-14B/D had a ceiling of 52,000 feet?