  Two novel topological black hole exact solutions with unusual shapes of
horizons in the simplest holographic axions model, the four-dimensional
Einstein-Maxwell-axions theory, are constructed. We draw embedding diagrams in
various situations to display unusual shapes of novel black holes. To
understand their thermodynamics from the quasi-local aspect, we re-derive the
unified first law and the Misner-Sharp mass from the Einstein equations for the
spacetime as a warped product $\M2 \times \Mco2$. The Ricci scalar $\Rhat$ of
the sub-manifold $\Mco2$ can be a non-constant. We further improve the
thermodynamics method based on the unified first law. Such a method simplifies
constructing solutions and hints at generalization to higher dimensions.
Moreover, we apply the unified first law to discuss black hole thermodynamics.
