Holographic entanglement entropy and complexity in St$\ddot{u}$ckelberg superconductor

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.