Corrected entropy of the rotating black hole solution of the new massive gravity using the tunneling method and Cardy formula

We study the AdS rotating black hole solution for the Bergshoeff-Hohm-Townsend (BHT) massive gravity in three dimensions. The field equations of the asymptotically AdS black hole of the static metric can be expressed as the first law of thermodynamics, i.e. $dE=TdS-PdV$. The corrected Hawking-like temperature and entropy of asymptotically AdS rotating black hole are calculated using the Cardy formula and the tunneling method. Comparison of these methods will help identify the unknown leading correction parameter $\beta_1$ in the tunneling method.