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First Law of Entanglement Entropy vs Black Hole Entropy in an Anomalous Theory

In this note we explore the validity of the first law of entanglement entropy in the context of the topologically massive gravity (TMG). We found that the variation of the holographic entanglement entropy under perturbation from the pure AdS background satisfies the first law upon imposing the bulk equations of motion in a given time slice, despite the appearance of instabilities in the bulk for generic gravitationl Chern-Simons coupling $μ$. The black hole entropy is different from the holographic entanglement entropy in a general boosted frame. This discrepancy however do not affect the entanglement first law.