Here's a top of the head response.
If G means the world as G knows it, then this will probably be in the next 80 years. G can change the odds on how it will end by becoming a couch potato, eating a fast food diet and taking up smoking. For an accidental death try Everest.
If G means the Earth, then perhaps what's more important is not when it ends, but when it stops being habitable.
Astronomers study stars by the light they give out. From measurements of a star's brightness, from the detailed spectrum of the light it emits and from how it seems to wobble around other stars nearby, you can work out it's mass, it's temperature, whether it's coming towards you or going away, how far away it is, what sorts of atoms there are in its outer atmosphere ... In particular, if you know how far away a star is you can say how long ago light must have left it and therefore how old it must have been (relative to the age of the Universe) when that light was given out. Knowing all these things means that we know a lot about different star types and their typical life cycles.
We know that big stars are brighter, hotter and have relatively short lives and that small stars are dimmer, cooler and live much longer. We also know that our star, the Sun, is a typical middle sized sort of star. It simply isn't big enough to explode in a Super Nova, or end it's life as a black hole or a neutron star. Instead, as it fuses it's nuclear fuel into heavier elements (in colloquial terms "burns up"), the pressure from the reactions inside will build up and it will swell. As it swells the surface will cool slightly and become redder. The Sun will be what's known as a Red Giant. After this phase it will spend a long time as a spinning cinder of hot rock known as a White Dwarf.
Astrophysicists think that in about 3 billion years the Sun will have expanded so much that it will completely smother Mercury and Venus and reach out almost as far as the Earth. Even though the Sun will be cooler, it will be so much closer, and cover so much of the sky, that all of the earth's water will evaporate and the conditions for life will cease to exist.
If G means the Universe, then the answer depends on whether of not it keeps expanding.
If it does, then eventually all of the stars, and the stars formed from the remnants of other stars, will have run out of fuel and will slowly cool down as they radiate their remaining heat energy into space. Meanwhile, if space keeps on expanding we'd begin to notice not only the space between galaxies expanding, as we do now, but also that between the atoms in molecules and even the particles within the atoms. Everything will be smeared out into an ever thinner smudge.
If it doesn't, and the Universe collapses back in on itself, it won't so much end as reset. A "new" Universe will bounce back out with no memory of the one we're in now.
This might take some time.
There'll still be rocks with Lichen when we're long gone
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