heat capacity
Heat capacity answers a simple question: how much heat do you have to pour into a lump of material to warm it up by one degree? A heavy ceramic mug or a firebrick soaks up a lot of heat and gives it back slowly, which is why it stays warm long after you fill it. Formally the heat capacity C is dQ/dT, the heat Q needed per degree of temperature rise T. Divide by mass and you get the specific heat (per gram); divide by moles and you get the molar heat capacity.
Where does that stored heat actually go? Into the jiggling of the atoms. In a ceramic the atoms are locked by strong ionic-covalent bonds into a rigid cage, and heat sets them vibrating about their sites. These coordinated vibrations are the lattice waves we call phonons. The Debye picture treats the crystal as a box full of these vibration modes: at high temperature every atom shares the heat equally and the molar heat capacity climbs to the classical Dulong-Petit value of about 3R, roughly 25 J per mole of atoms per kelvin (R is the gas constant, 8.314 J/mol/K). At low temperature the modes freeze out one by one and the heat capacity falls off as T^3. The changeover is set by the Debye temperature: stiff, light-atom ceramics like diamond, BeO and SiC have very high Debye temperatures, so at room temperature they are still on the rising part of the curve and their heat capacity per gram is comparatively low.
Heat capacity matters wherever heat must be stored, moved, or survived. Refractory linings and kiln furniture with high heat capacity waste fuel every time you heat them up and cool them down, so lightweight low-mass insulation is preferred for cycling furnaces. In thermal shock it is the amount of heat a surface must dump to cool that helps set the temperature gradient. And because thermal conductivity in ceramics is essentially heat capacity times phonon speed times a mean free path, you cannot understand how a ceramic conducts heat without first understanding how it stores it.
Alumina has a specific heat near 0.78 J/g/K at room temperature, rising toward 1.2 J/g/K near 1000 degrees C. Its molar mass is about 102 g/mol for Al2O3 with 5 atoms, so 3R times 5 is about 125 J/mol/K, or 1.22 J/g/K — the classical Dulong-Petit ceiling that the high-temperature value approaches.
The Dulong-Petit rule of 3R per mole of atoms sets the high-temperature ceiling that most crystalline ceramics approach.
Heat capacity per mole of atoms is remarkably similar across crystalline ceramics at high temperature (near 3R), so wide differences in specific heat per gram mostly reflect differences in atomic weight, not in some special heat-storing power.