Imagine two objects, identical and having the same mass. Now my question is, if you heat one object and do not do anything with the other object, will the hot object possess more gravity (because it possesses more energy and energy is interchangeable with mass).
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2Related, though surprisingly I can't find an exact duplicate, Will heating up an object increase its mass? and the links therein. – John Rennie Mar 17 '15 at 06:55
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To see this start with an isolated cold object. If we make it spherical then the spacetime curvature around it will be (approximately) described by the Schwarzschild metric, which is nice and simple.
Now to make the object hot we have to add something to it. We could shine in radiation or fire in objects with high kinetic energy, but whatever we do is necessarily going to increase the amount of mass/energy the object has. The stress-energy tensor doesn't distinguish between matter and energy - both act as sources for the gravitational field.
So the answer is (as you had guessed) that heating the object will increase the gravitational field around it.
John Rennie
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@ChrisWhite: a real object is neither time independant nor the only object in the universe – John Rennie Mar 17 '15 at 07:49