This is a fairly complex question, one that cannot easily be answered in a quick response.
At one time, when cartridges were fairly similar and used very similar bullets, it was relatively easy to come up with simple, broad statements that would be valid in most situations. For example, it really didn't much matter if your used a 308, a 270, a 30-30, or a 30-06.... You went to Walmart, bought a box of jacketed soft points by Remington, Winchester or Federal and called it good. If it generated 1000 or 1200 ft/lbs of energy or more at impact, you were good to go on deer....
This doesn't hold true as much anymore. Its far more complex...
Take the .300 Blackout. This cartridge would be a complete non-starter on deer using the criteria outlined above. Its nota viable candidate using old school bullets and old school technology/loads. However, loaded with Barnes 110 TSX bullets, it suddenly works far better than it should...
Need another example? .223 Rem. Circa 1980, with the then typical 55 basic jacketed soft point and its decidedly iffy as a deer killer. Same cartridge, same velocities, same energy but loaded with newer bullets and it starts acting far better than it should..
Generally, on deer, using basic unsophisticated bullets, the old 1000 ft'lbs of energy is a good absolutely minimum. Of course, more is better. However, there are literally dozens of exceptions now, thanks to improved technologies. Things like bonded bullets take a kind of sort of minimal cartridge and give it a good boost in performance.
As a rule, it's very important to match the BULLET to the game AND the cartridge. At one time 99% of cartridges launched a cup and core jacketed soft point bullet somewhere between 2500 and 2900 fps. It was hard to go wrong. A 150 Win powerpoint in 308, a 130 in 270 or a 180 in 30-06 worked well. Now we have very very fast cartridges, easily capable of launching bullets to 3200 fps +++. Regular cup and core bullets do not do well at those speeds. Monolithics like Barnes TTSX do very well. Nosler Ballistic Tips are total failures.
Turn the tables. Look at something slow, like 6.5 Grendel. A heavy 140 Barnes TTSX is a miserable choice. Its too slow. A Nosler Ballistic Tip is a FAR better choice at 2300 fps muzzle velocity. In most cases, bullets like the Barnes TTSX totally suck at low speeds..... BUT Barnes makes a specific-for-the 300 Blackout 110 gran TSX and its MARVELOUS in that application....
See what I mean? its really really complex.
Sorry, there isn't a really good rule of thumb anymore. Personally, 1200 ft lbs is still a good rule. There are exceptions. If this is something of real interest, don't spend all your time thinking about cartridges and spend a fair bit of time learning about the specific bullets on the market. EVERY BULLET, every single bullet, has a VERY specific operational "window" or sweet spot that it will work very well at. Slower or faster than that sweet spot and it will not do well....