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We analyze the dependence of the effective action and the entanglemententropy in the Maxwell theory on the gauge fixing parameter $a$ in $d$dimensions. For a generic value of $a$ the corresponding vector operator isnonminimal. The operator can be diagonalized in terms of the transverse andlongitudinal modes. Using this factorization we obtain an expression for theheat kernel coefficients of the nonminimal operator in terms of thecoefficients of two minimal Beltrami-Laplace operators acting on 0- and1-forms. This expression agrees with an earlier result by Gilkey et al. Workingin a regularization scheme with the dimensionful UV regulators we introducethree different regulators: for transverse, longitudinal and ghost modes,respectively. We then show that the effective action and the entanglemententropy do not depend on the gauge fixing parameter $a$ provided the certain($a$-dependent) relations are imposed on the regulators. Comparing theentanglement entropy with the black hole entropy expressed in terms of theinduced Newton's constant we conclude that their difference, the so-calledKabat's contact term, does not depend on the gauge fixing parameter $a$. Weconsider this as an indication of gauge invariance of the contact term.

Remarks on effective action and entanglement entropy of Maxwell field in generic gauge