When will the Final Ice Age happen?
At the moment, nobody knows for sure whether or not changes in the Sun's luminosity can cause long-term climate changes. There have been various studies done, most since the advent of global warming, that have tried to address this. The conclusions have been interesting:


The Sun is not responsible for the recent warming of the Earth.
It's not yet known whether or not the Sun has influenced climate in the past.


Here are some studies (one supporting each side of the long-term debate):


A report by the National Research Council (available here) found that sunspot activity could be a major contributor to small changes in the Sun's brightness. There is a periodic trend in sunspot activity that lasts 11 years, and studies have considered this when looking at long-term trends. One target of inquiry is the Maunder Minimum, a period of low sunspot activity:



The Maunder Minimum coincided with the Little Ice Age, a short period of low temperatures. In fact, the Sun may be entering a similar period within the next few decades. Observations could show whether or not there is a link between the two events.

However, the NRC concluded that it is hard, if not impossible, to find any data supporting theories that there is a long-term patter. Hal Maring summarized it best:


  Hal Maring, a climate scientist at NASA headquarters who has studied the report, notes that “lots of interesting possibilities were suggested by the panelists.  However, few, if any, have been quantified to the point that we can definitively assess their impact on climate.”

In the paper Possible solar origin of the 1,470-year glacial climate cycle demonstrated in a coupled model by Holger et al., periods called Dansgaard–
Oeschger events were studied. These are climate oscillations lasting 1,470 years. Holger et al. analyzed ice cores from Greenland and attempted to model solar forcing, the amount of energy absorbed by Earth or radiated into space. They also used the ocean and atmosphere to account for changes (the NRC report, too, concluded that solar effects can trickle down through the atmosphere and other parts of Earth). Their conclusion was that, despite the fact that no records of 1,470-year solar cycle exist, the analysis supported a connection between solar variability and changes in Earth's climate.

Before anyone gets too excited, I'll add this: 1) I haven't been able to find any comments from other scientists on the paper, so I'm not sure how well-received it was, and 2) The 1,470-year cycle can not account for long-term glacial periods, or the recent rise in global temperatures.


Another potentially interesting (and paywalled) paper can be found here. An informative website is here, but I'm not sure if it's entirely neutral. It does seem to present both sides, though.

I'll end on this: Whether or not the Sun severely impacts the Earth's climate, Milankovitch cycles are the biggest influence on ice ages.



It will take some pretty drastic changes in the Sun's luminosity to change the glacial periods . . . which the Sun will be undergoing in about billion years or so. It will still be on the main sequence, and will stay there for 4.4 billion years more, but it will have brightened by about 10%.

Once of the consequences of this will be that the Earth will lose its oceans. Through a process called the "moist greenhouse" effect, water heats up and reaches the atmosphere, where the hydrogen atoms are freed from the oxygen atoms and escape to space. This occurs at a tiny, tiny rate at present, but in about one billion years, the Sun will be bright enough that this will occur at a catastrophic rate.

That places the last glacial period somewhere between now and one billion years in the future. To figure this out, we have to look at long term patterns, such as Milankovitch Cycles. This graph illustrates them best:



From the present day to the far right, though, is only 800,000 years. That's unhelpful. This graph, while a bit clearer, shows only the variations without accounting for any changes in the Sun's luminosity (though that will not affect us during the next 800,000 years!).



Here's the actual answer to your question: We don't know. That's because some parts of the Earth will be affected more by the Sun's brightening that others. For example, the NRC found that the Pacific Ocean experienced a strong relationship with solar activity. So as the Sun gets brighter, some areas could still be strongly affected by Milankovitch cycles, while others might find themselves getting hotter than they would normally be.

We can be sure of one thing, though: It will certainly be before one billion years from now!