AX J0049.4−7323 is a Be/X-ray binary in the Small Magellanic Cloud hosting a ~750 s pulsar which has been observed over the last ~17 years by several X-ray telescopes. Despite numerous observations, little is known about its X-ray behaviour. Therefore, we coherently analysed archival Swift, Chandra, XMM-Newton, RXTE, and INTEGRAL data, and we compared them with already published ASCA data, to study its X-ray long-term spectral and flux variability. AX J0049.4−7323 shows a high X-ray variability, spanning more than three orders of magnitudes, from L ≈ 1.6 × 10³⁷ erg s⁻¹ (0.3−8 keV, d = 62 kpc) down to L ≈ 8 × 10³³ erg s⁻¹. RXTE, Chandra, Swift, and ASCA observed, in addition to the expected enhancement of X-ray luminosity at periastron, flux variations by a factor of ~270 with peak luminosities of ≈2.1 × 10³⁶ erg s⁻¹ far from periastron. These properties are difficult to reconcile with the typical long-term variability of Be/XRBs, traditionally interpreted in terms of type I and type II outbursts. The study of AX J0049.4−7323 is complemented with a spectral analysis of Swift, Chandra, and XMM-Newton data which showed a softening trend when the emission becomes fainter, and an analysis of optical/UV data collected by the UVOT telescope on board Swift. In addition, we measured a secular spin-up rate of Ṗ = (−3.00 ± 0.12) × 10⁻³ s day⁻¹, which suggests that the pulsar has not yet achieved its equilibrium period. Assuming spherical accretion, we estimated an upper limit for the magnetic field strength of the pulsar of ≈3 × 10¹² G.