We consider the refinement of the holographic entanglement entropy for the holographic dual theories to the AdS solitons and AdS black holes, including the corrected ones by the Gauss-Bonnet term. The refinement is obtained by extracting the UV-independent piece of the holographic entanglement entropy, the so-called renormalized entanglement entropy which is independent of the choices of UV cutoff. Our main results are (i) the renormalized entanglement entropies of the AdS$_{d+1}$ soliton for $d=4,5$ are neither monotonically decreasing along the RG flow nor positive definite, especially around the deconfinement/confinement phase transition; (ii) there is no topological entanglement entropy for AdS$_5$ soliton even with Gauss-Bonnet correction; (iii) for the AdS black holes, the renormalized entanglement entropy obeys an expected volume law at IR regime, and the transition between UV and IR regimes is a smooth crossover even with Gauss-Bonnet correction; (iv) based on AdS/MERA conjecture, we postulate that the IR fixed-point state for the non-extremal AdS soliton is a trivial product state.
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