The paper derives an exact static Kaluza-Klein black hole with four electric charges and squashed S3 horizon in 5D STU-W2U supergravity and verifies that its thermodynamics obey the first law and Bekenstein-Smarr formula when the compact dimension length is treated as a thermodynamic variable.
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Static four-charge squashed black hole in five-dimensional $STU-W^2U$ supergravity and its thermodynamics
The paper derives an exact static Kaluza-Klein black hole with four electric charges and squashed S3 horizon in 5D STU-W2U supergravity and verifies that its thermodynamics obey the first law and Bekenstein-Smarr formula when the compact dimension length is treated as a thermodynamic variable.