What properties can be changed to maximize a p-n diodes effectiveness at higher temperatures?
There are two main principles that allow a diode to go into breakdown:


Avalanche breakdown: When the electric field in the depletion region gets too strong, charges can gain enough kinetic energy to generate more electron-hole pairs. These new charges can potentially gain enough kinetic energy to excite other electron-hole pairs. It causes a multiplication effect, called avalanching. It should be noted that if no charges are present inside the depletion region, then the diode will also not avalanche. However, a single charge can disrupt that. This effect is exploited in for example Single-Photon Counting, where a single photon can generate an electron-hole pair, starting a surge of current that can be measured.
Zener breakdown: It is possible that at the P-N junction, the conduction band of the N-region gets very close to the valence band of the P-region (see image on the right - Wikipedia).



This breakdown happens usually at high doping levels. It is by my knowledge mainly exploited by Zener diodes for lower Zener-breakdown voltages.


It should be noted that Zener breakdown happens faster for increasing temperature as more charges become excited (also reducing the band gap this way) that can tunnel. Avalanche breakdown happens later, as the heat will increase the number of collisions of moving charges, making it harder to build up kinetic energy.

In both cases, reducing the doping levels will reduce breakdown effects. For avalanche, the depletion width will increase, causing the electric field to decrease (\$|E| \sim \Phi / w\$). The built-in potential will decrease as well, leading to a lower internal potential. Zener breakdown will also be reduced, as the built-in potential decreases.

Another way of reducing breakdown effects is by increasing the depletion width in other ways. For example, you can apply a smooth doping profile, or move to a PIN diode altogether.