<p>We present a detailed study of the regularized on-shell action of the recently found asymptotically AdS₇ black hole supergravity solution with three angular momenta and two electric charges. We show that a particular choice of finite counterterms in the holographic renormalization procedure yields a result for the on-shell action of the supersymmetric limit of the black hole solution which is in perfect agreement with the large <i>N</i> limit of the superconformal index of the dual 6d 𝒩 = (2, 0) <i>A</i>_<i>N</i> SCFT on <i>S</i>¹ × <i>S</i>⁵. We also discuss a generalization of this 7d supergravity background to a new general family of black holes with horizons given by the <i>L</i>^{<i>p</i>,<i>q</i>,<i>r</i>} 5d Sasaki-Einstein manifolds. The regularized on-shell action of these new 7d black holes is also in agreement with supersymmetric localization results for the 6d 𝒩 = (2, 0) SCFT on <i>S</i>¹ × <i>L</i>^{<i>p</i>,<i>q</i>,<i>r</i>}. Finally, we briefly discuss a generalization of known AdS₅ gauged supergravity backgrounds to new 5d black holes with <i>L</i>(<i>p</i>, <i>q</i>) lens space horizon topologies.</p>