Shouldn't a cold object feel less gravity?

Question

The premises for my question are,

Gravity isn't a force but curved spacetime

"If gravity isn't a force, how does it accelerate objects?" (complicated answer)

If the answer below, to the question linked above, is correct, I feel it's easier to understand:

"Even objects "at rest" (in a given reference frame) are actually moving through spacetime, because spacetime is not just space, but also time: apple is "getting older" - moving through time."

Time is the result of causality -- to my understanding this means that something ages because of the interactions of its smaller parts, and at the speed of light for example, those interactions don't happen hence time doesn't pass.

So, a very cold object would move less through time (because its internal parts are moving slower, which means less causal interactions), and hence "feel less gravity".

Answer 1

It is true that gravity acts on things that have mass and things that possess energy. But the energy content of mass is gigantic for macroscopic objects via $E = mc^2$. On the other hand, the difference in internal energy content of a massive object at two temperatures goes like $\Delta E = mC_p \Delta T$ where $C_p$ is the specific heat of the material. This amount of energy will create an immeasurably small amount of gravity in comparison.