The adum trials are challenges that must be completed within Temple Grounds in order to gain access to the High Prophet's Room. The trials start at the top left warp gate, the Breath of adum.
Once inside go below 30 speed until you reach the other side.
Once you enter the next galaxy, you need to spell adum with lowercase letters and the gate will open up. Then, once you are in the next room you need to get to the other side of the warp as quickly as possible before you die.
Lastly, the next room a key will appear on the sun which you will need to complete the mission. If it doesn't appear immediately, warp back into the previous galaxy to wait for it to spawn.
A Fitting Reward
Mission to get a Divine Behemoth.
Mission starts at the hidden base within the Temple Grounds. Requires A Fitting Reward to start.
The secret task is done by going through the adum Trials a second time.
Spell the word "base" instead of "adum" but he wont let you know each letter is correct (like he did before with "adum"), but just spell "base" and only when you finished that word, he tells you its done and access to the base is given.