Time-dependent and constituent-specific spectral changes in soft near edge X-ray spectroscopy (XAS) of an [Fe/MgO]$_8$ metal/insulator heterostructure upon laser excitation are analyzed at the O K-edge with picosecond time resolution. The oxygen absorption edge of the insulator features a uniform intensity decrease of the fine structure at elevated phononic temperatures, which can be quantified by a simple simulation and fitting procedure presented here. Combining X-ray absorption spectroscopy with ultrafast electron diffraction measurements and ab initio calculations demonstrate that the transient intensity changes in XAS can be assigned to a transient lattice temperature. Thus, the sensitivity of transient near edge XAS to phonons is demonstrated.