  High-temperature high-pressure superconducting hydrides are typically
characterized by cage-like hydrogenic lattices filled with electropositive
metal atoms. Here, density functional theory based evolutionary crystal
structure searches find two phases that possess these geometric features and
are related to the Type-II clathrate structure. In these $Fd\overline{3}m$
Li$_2$CaH$_{16}$ and $R\overline{3}m$ Li$_2$CaH$_{17}$ phases the calcium atom
occupies the larger cage and the lithium atom the smaller one. The highest
superconducting critical temperatures predicted within the isotropic Eliashberg
formalism, 330~K at 350~GPa for $Fd\overline{3}m$ Li$_2$CaH$_{16}$ and 370~K at
300~GPa for $R\overline{3}m$ Li$_2$CaH$_{17}$, suggest these structures fall in
the class of high-energy-density quantum materials known as hot
superconductors. As pressure is lowered the cage-like lattices distort with the
emergence of quasimolecular hydrogenic motifs; nonetheless Li$_2$CaH$_{17}$ is
predicted to be superconducting down to 160~GPa at 205~K.
