From The Nitrosyl Azide Potential Energy Hypersurface: A
High-Energy-Density Boom or Bust? J. Am. Chem. Soc. 1996, 118, 4860-4870 :
[Abstract] ...While the ring isomer is predicted to be the most stable structure on the
hypersurface, the barrier to dissociation is most likely between 1 and 2 kcal mol-1 (including zero-point vibrational
energy [ZPVE], the existence of any barrier becomes questionable) making isolation theoretically possible but
experimentally difficult. This small barrier also detracts from the attractiveness of the N4O ring structure as a high energy-density material. The trans-chain isomer, however, lies in an energy valley with higher sides, consistent
with its previous experimental observation.
[full text]...In an ideal five-membered ring
with 6 $\pi$ electrons, the $\pi$ electrons would be distributed evenly
among all the bonds. In the present case, the highly electronegative
O atom prefers to keep electrons around itself, leading
to a partial negative charge on the oxygen. Energetically, there
is a certain degree of stability associated with the ring isomer,
although not on the order of common aromatic systems. The
ring isomer is predicted to be at most (DZP CISD) 20.9 kcal
mol-1 more stable than the trans-chain isomer; however, this
value decreases to 13.2 kcal mol-1 with TZ2P CCSD.
The ring isomer TS to dissociation into N2 and N2O is shown
in Figure 5. ... Energetically, the barrier to dissociation
is at most 15.3 kcal mol-1 (DZP CISD) and drops lower
with improvements in both basis set and correlation scheme.
In going from a DZP to a TZ2P basis set, for example, this
barrier drops by 5.2 kcal mol-1 for CISD and 4.3 kcal mol-1
for CCSD. Assuming a similar trend in moving from DZP
CCSD(T) to TZ2P CCSD(T), the ring dissociation barrier is
expected to drop to 1-2 kcal mol-1 with the addition of f-type
functions possibly making it even lower. A barrier of this size
lies below the ZPVE, throwing doubt on the existence of the
N4O ring isomer.
So thermodynamically, yes, the ring form is the lowest energy isomer. However, all the isomers are unstable to decomposition and the linear isomer is in a deeper potential energy well than the ring isomer. Therefore, the linear isomer is easier to observe experimentally for kinetic reasons.
There may be additional information in Theoretical study on structures and stabilities of N4X (X = O, S, Se, Te) series International Journal of Quantum Chemistry Volume 109, pages 226–235.