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In this note, the off-shell generalized Abbott-Deser-Tekin (ADT) formalism isapplied to explore the mass of Kerr-anti-de Sitter (Kerr-AdS) black holes invarious dimensions within asymptotically rotating frames. The cases in four andfive dimensions are explicitly investigated. It is demonstrated that theasymptotically rotating effect may make the charge be non-integrable orunphysical when the asymptotic non-rotating timelike Killing vector associatedwith the charge is allowed to vary and the fluctuation of the metric isdetermined by the variation of all the mass and rotation parameters. To avoidsuch a dilemma, we can let the non-rotating timelike Killing vector be fixed orperform calculations in the asymptotically static frame. Our results furthersupport that the ADT formalism is background-dependent.

A note on the mass of Kerr-AdS black holes in the off-shell generalized ADT formalism