[Cobalt-Blue (OP)]

Could a "hot Jupiter" or "super earth" exist in an orbit 1 AU from a star like our own, and if so, could an Earth size planet be in its L4 or L5 position without being tidally locked to either the planet or the star?

[syhprum]

The blackbody temperature at 1 AU is about 250°K what would keep a hot Jupiter hot ?

[evan_au]

"hot Jupiter"

This term is generally reserved for large planets (the size of Jupiter) but in orbits very close to their star (0.015-0.5AU), with radiation from their star being much stronger than what we receive on Earth.

But I guess a Jupiter-sized planet in Earth's orbit (1AU) is not so far from the usual upper limit (0.5 AU). It is certainly a long way from where astronomers expected gas giants like Jupiter and Saturn to form (and they presumed, to stay there, once they were formed).

Could a "hot Jupiter" or "super earth" exist in an orbit 1 AU from a star like our own?

Yes. Astronomers were surprised to see them so close to their parent star. But if they can get that close, they could certainly be in a 1 AU orbit.

if so, could an Earth size planet be in its L4 or L5 position without being tidally locked to either the planet or the star?

Tidal friction is quite weak at the distance of the L4 & L5 points (the time to become tidally locked increases as the 6th power of the orbital distance). If the Earth has not become tidally locked to the Sun by now, it would not become tidally locked to a Jupiter-sized planet at the same distance.
See: https://en.wikipedia.org/wiki/Tidal_locking#Timescale

Of more concern is the stability of the L4 and L5 points. The strength of the tidal interaction is fairly weak, and disturbance from other planets can set up a resonance in the body at the L4 & L5 point which breaks it out of this stable zone.

Being in the same orbit as a Jupiter-sized body is a perilous place to be - the Earth-sized moon could crash into the planet, be flung out of the planetary system, into the star, or an unbearably hot or cold orbit.

So you would have to assume that there are no other large planets in close orbits which could disturb the delicate balance in the L4 & L5 position. Small rocky planets in distant orbits would not disturb the L4/L5 positions of a Jupiter-sized planet excessively.

The blackbody temperature at 1 AU is about 250°K what would keep a hot Jupiter hot ?

For an extrasolar Hot Jupiter: An orbit of 0.015 AU (well inside Mercury's orbit) produces an incredibly high black-body temperature.

For the real Jupiter in our Solar System: Leftover heat from its formation. Radioactive decay from a rocky core. Potentially a tiny amount of Hydrogen fusion in its Hydrogen core. The bigger the body, the longer it holds its heat (all other things being equal).

The Real Jupiter has a "cloud-top" temperature of about 340K, which is well above the black-body temperature at it's icy orbit in the Solar System.
See: https://en.wikipedia.org/wiki/Jupiter#Internal_structure

[Cobalt-Blue (OP)]

Thank you for the information.

[Professor Mega-Mind]

Gentlemanses ,
Beautiful as "Earths" around Jupiters might appear on screen , they would be a dangerous night-mare , even if the Universe made them , which it essentially doesn't .  The gas-giant proto-planetary disk is not nearly massive enough to form an Earth , but if it did , tide-induced volcanism would be a major problem .  Even worse , a max. gas-giant at 1 AU would have a monstrous magnetic field , containing thousands of times more radiation than Jupiter's . Talk about lethal !  Earth around red-dwarf star is so much nicer !  Very picturesque , and yet you live !
Tatooine , yeah !
P.M. 
**Links/references @ Reply#7 .
》Note : As mentioned above , the L-4&5 Points are too unstable to contain a planetary body for long .
*Addendum : An Earth-like planet orbiting a Jupiter-class planet , would likely have a much larger tidal-bulge than Earth does . This would cause a very rapid rotational slow-down , ending with tidal-locking . Days on such planets would be from several days to several weeks long .

[Halc]

Could a "hot Jupiter" or "super earth" exist in an orbit 1 AU from a star like our own, and if so, could an Earth size planet be in its L4 or L5 position without being tidally locked to either the planet or the star?

I think they've spotted a Jupiter size planet well inside the orbital radius of Earth, so it does happen that one forms that close.

Tidal locking issue would not be that much different.  The forces acting on Earth would be about the same as they are now, so if Earth spins now, it would still be spinning in that setup.  Earth has the moon which acts as a much larger brake to its spinning than does the sun or the effect from this hot-Jupiter.

[Professor Mega-Mind]

...............The Alternative.
True that . I was accepting the fact established above , that L-4/5 are unstable for an Earth .  I addressed  the only stable possibility ; Endors . An Earth-class planet would cause negligible braking-effect for a max. Jupiter-class planet .  Therefore , the Earth would be lifted into such a high orbit , that it would be at risk of separation from the planetary system .  If this happened , later collision with the gas-giant would be likely .  If it didn't , absurdly powerful radiation would continue to scour the planet .
Addendum : Jupiter is ~50 times wider than Luna . The gravitational force experienced at Io is ~5 times that experienced at Luna . The 10-hr. Jovian rotation drags it's tidal-bulge much further sideways than the Terran .bulge , thus increasing the draw-angle and the sideways-pull induced by Jupiter's bulge . The tidal-forces experienced by a satellite "Terra" would be hundreds of times stronger than those experienced by Luna . Tide-lock and tidal-lifting would occur quickly .
Alright , Planet Yavin-4 debunked !
P.M.   .
https://phys.org/news/2020-06-titan-migrating-saturn-faster-previously.html   

[Professor Mega-Mind]

...........The Rare Occurance .
One extremely rare occurance , that can give rise to a "Pandora" world , is the "glancing-collision/capture" scenario .  IF a gas-giant planet migrates all of the way in to an Earth's orbit , a rocky-planet/gas-giant versus gas-giant collision , could result in a Triton/Luna-style capture by the gas/ice giant . A Neptune-class giant planet would have a much more benign radiation environment  than a Saturn or Jupiter class one . The gas-giant would likely slow it's rotation quickly , being only 20 times as massive , and yield much weaker tidal forces upon the Earth-like planet , as well .  For any stellar size above red-dwarf , the star/planet tidal-forces should be small enough to avoid destabilization of the Earth-planet's orbit .  This for the life of the parent star . 
*No planet Yavin , only Neptune .
P.M.
Note - A jovian-class planet is more likely to shred/absorb a grazing impactor , than an ice-giant planet .  This is due to it's higher density and gravity .
Note 2 - Impact MUST occur in a warm zone , so as to drive off the volatile-elements . Further out , and the new moon will aquire a permanent ice/water shell .
》For reference , link to NSF thread : How do proto-planets form ?
www.thenakedscientists.com/forum/index.php?topic=74806.new;topicseen#new 
*Update by me available on Quora thread : www.quora.com/Hypothetical-Scenario-If-Jupiter-took-the-place-of-Earth-and-Earth-became-a-moon-of-Jupiter-could-we-survive?
>Read my comments to .OP. Dave Consiglio , especially the June 1 entry at the very end of comments.
*Separate link below :
www.quora.com/Is-it-possible-to-find-an-inhabitable-moon-anywhere-in-the-universe-or-does-it-need-terraforming?   
**.Bonus Ref.: quora.com/Do-habitable-exomoons-exist-1/answer/Derek-Hendricks-7?ch