Sempiternal, Æviternal, Omniëssent, & Omnitemporal
It is not clear why eternal or everlasting wouldn’t be good enough as is, but if you want a higher-falutin’ term, the most obvious answer is sempiternal, which the OED defines as:
Enduring constantly and continually; everlasting, eternal.
That takes the Latin semper meaning “ever” or “always” and combines it with ‑ternal to produce a fancy word meaning “everlasting”.¹
The OED also attests the adjective ever-being with the sense “that which always is”, although the resulting noun everbeingness the OED considers obsolete.
A rather less common word with the same meaning as sempiternal is aeviternal (also spelt æviternal), which the OED defines as:
Everlasting, endless, eternal.
This time it comes from aevum meaning “an age”, which is related to aeon (sometimes spelt æon or eon) and thence to Greek αἰών.
This is related to but different from longeval (also spelt longaeval or longæval), which just means long-lasting or long-lived, not ever-lasting or ever-living. But you can see the same aevum root at work, this time at the end of the word instead of at its beginning.
The word omniessent (also spelt omni-essent or omniëssent) that was kindly suggested by Janus in a potentially ephemeral² comment is simply the adjective corresponding to the noun omni-essence, which the OED defines as:
universal essence or being
To me omniessent more suggests ubiquity of being than it does omnitemporaneity of the same.
Speaking of omnitemporaneousness, that brings me to my final entry, omnitemporal, defined by the OED as:
relating to all times; including in its meaning all the various tenses
Summary: Native or Fancy?
These are all long and highly fancy — perhaps even fanciful — words built of Latinate roots from antiquity instead of being built of native roots the way everlasting and everbeing are.
As such, they might have a place in a lofty register such as that of Aquinas, but for more casual ones, simpler words are likely better.
- This is like how sempervirent is a fancy word that means “evergreen”, visible in the taxon for the coast redwood, Sequoia sempervirens.
One should take some care not to confuse sempervirent with sempervivent, where the latter means “everliving” or “lives-forever”, a term once applied to the succulent we call hen-and-chicks or houseleek and which survives as the genus Sempervivum.
- Where ephemeral and eternal are antonyms.