It seems clear to me that the object at rest has no energy.
I'm not sure there is such a thing as "an object at rest."
An object at rest, absolute rest, would also have 0 Kelvins of energy. I do not mean 0 degrees Kelvins (although both refer to almost the same thing), I mean Kelvins, as in molecular energy. At 0 Kelvins, matter would have absolutely no molecular energy. Or any other kind of energy.
However, as GOF said, in a round about way, there is no such thing as an object at rest. Everything contains energy, whether it is molecular, nuclear, potential, etc., etc. Even at the Zero Point.
A very good example is Helium. Even at 0 degrees Kelvin, also viewed as being the Zero Point, helium is still a liquid. This means, that even at the Zero Point, helium still contains energy, enough to keep it liquid. That is until we also apply pressure. It takes about 4 atmospheres of pressure to solidify helium at 0 degrees Kelvin. This kind of proves (but not truly) that even at the Zero Point, there is still energy. Thus, there is no such thing as "an object at rest."
But! It would be interesting if such a thing could be achieved through technology. That would mean an object at "absolute rest" would literally become The "center of the universe" since it would be at absolute rest and everything else is moving away from it.
The speed of gravity is infinite. As GOF pointed out, the effects of an object's attractive force (gravity), no matter how miniscule, is determined by where it
IS, not where it
WAS. So imagine a true, pure, hypergravitic drive. You could literally put yourself anywhere in the universe in less than an instant. No real time passing. Currently, in my FtF SM campaign, that is a type of drive the Terran Star Empire is currently working on. However, the best they have been able to do is about 10 to 15 LY in (as they call it) an instant. And, they have had problems with the X-Ships either imploding or exploding sometimes at the destination point. Currently, they are using a much less advanced system known as the Hypershunt Jump Drive, which I have mentioned here. I just have not mentioned that it works on hypergravitics which was first introduced by Isaac Asimov in his Foundation universe. Basically, that drive shunts the ship to FTL velocities with no real time passing. The deshunt also occurs with no real time passing. However, time still normally passes while in the hypershunt. And, no, for some reason, time does dilate while in the hypershunt. Time still passes normally. In other words, if you spent five days (432,000 seconds) in the hypershunt, then five days (432,000 seconds) has passed outside the hypershunt. Most often, the Terrans simply refer to such as "shunt" and "deshunt."
Once advanced enough, the Terran Star Empire will be able to "shunt" their ships from location to location with no real time passing, or if you prefer, instantaneously. Then as they continue to advance, they will discover how that drive will literally have no distance limitations. However, that is quite a bit of time down the stream.
rmfr