Improper Ferroelectricity and Piezoelectric Responses in Rhombohedral ($A$,$A^{prime}$)$B_2$O$_6$ Perovskite Oxides

High-temperature electronic materials are in constant demand as the required operational range for various industries increases. Here we design $(A,A^prime)B_2$O$_6$ perovskite oxides with [111] ``rock salt $A$-site cation order and predict them to be potential high-temperature piezoelectric materials. By selecting bulk perovskites which have a tendency towards only out-of-phase $B$O$_6$ rotations, we avoid possible staggered ferroelectric to paraelectric phase transitions while also retaining non-centrosymmetric crystal structures necessary for ferro- and piezoelectricity. Using density functional theory calculations, we show that (La,Pr)Al$_2$O$_6$ and (Ce,Pr)Al$_2$O$_6$ display spontaneous polarizations in their polar ground state structures; we also compute the dielectric and piezoelectric constants for each phase. Additionally, we predict the critical phase transition temperatures for each material from first-principles to demonstrate that the piezoelectric responses, which are comparable to traditional lead-free piezoelectrics, should persist to high temperature. These features make the rock salt $A$-site ordered aluminates candidates for high-temperature sensors, actuators, or other electronic devices.