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 ~ 1.6E37 erg/s (0.3-8 keV, d=62 kpc) down to L ~ 8E33 erg/s. 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 ~2.1E36 erg/s 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 $dot{P}=(-3.00pm0.12)times 10^{-3}$ 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 ~3E12 G.
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