  We analyze the holographic entanglement entropy in a soliton background with
Wilson lines and derive a relation analogous to the first law of
thermodynamics. The confinement/deconfinement phase transition occurs due to
the competition of two minimal surfaces. The entropic c function probes the
confinement/deconfinement phase transition. It is sensitive to the degrees of
freedom (DOF) smaller than the size of a spatial circle. When the Wilson line
becomes large, the entropic c function becomes non-monotonic as a function of
the size and does not satisfy the usual c-theorem. We analyze the entanglement
entropy for a small subregion and the relation analogous to the first law of
thermodynamics. For the small amount of Wilson lines, the excited amount of the
entanglement entropy decreases from the ground state. It reflects that
confinement decreases degrees of freedom. We finally discuss the second order
correction of the holographic entanglement entropy.
