  The holographic superconductors, as one of the most important application of
gauge/gravity duality, promote the study of strongly coupled superconductors
via classical general relativity living in one higher dimension. One of the
interesting properties in holographic superconductor is the appearance of first
and second order phase transitions. Recently, another active studies in
holographic framework is the holographic entanglement entropy and complexity
evaluated from gravity side. In this note, we study the properties of the
holographic entanglement entropy and complexity crossing both first and second
order phase transitions in St$\ddot{u}$ckelberg superconductor. We find that
they behave differently in two types of phase transitions. We argue that
holographic entanglement entropy and complexity conjectured with the volume can
also be a possible probe to the type of superconducting phase transition.
