What evidence is there for additional dark energy coming into existence when space increases? As I understand since cosmological constant is a 'constant' - increasing the space must generate additional dark energy that fills that space
This does sound counter-intuitive to me, but I am sure someone has looked at this and tested this thoroughly before making a claim that energy conservation does not hold.
I couldn't find any articles on how it was measured and confirmed that cosmological constant is a constant, and not that it changes in expanding universe.
 A: At the moment we don't know what dark energy is so we formulate hypotheses and compare them to the experimental data. The two most popular hypotheses are:


*

*dark energy is due to a cosmological constant

*dark energy is a scalar field referred to as quintessence
and both of these have the property that the total energy inside a volume of space increases as that volume of space expands. The experimental evidence is consistent with both hypotheses (with some restrictions on the time dependance of the quintessence field) so the current position is that we cannot rule out non-conservation of energy.
I should note that views differ on whether or not energy is conserved in an expanding universe. It depends on what you include when calculating the energy.
It's not obvious how you'd prove or disprove the point. If you could come up with a theory that did not create energy and show it was a better fit to the evidence than a cosmological constant or quintessence then that might do it. However at the moment the experimental data is not good enough to do this.
A: 
What evidence is there for additional dark energy coming into existence when space increases?

None. 

As I understand since cosmological constant is a 'constant' - increasing the space must generate additional dark energy that fills that space

That's what people say. But we have no evidence whatsoever of anything wherein conservation of energy does not apply. You can find people saying "energy is not conserved in GR", but others dispute this. 

This does sound counter-intuitive to me, but I am sure someone has looked at this and tested this thoroughly before making a claim that energy conservation does not hold.

Don't be sure. There is no actual scientific evidence to support the claim.  

I couldn't find any articles on how it was measured and confirmed that cosmological constant is a constant, and not that it changes in expanding universe.

There aren't any. But have a google on fine structure constant variation. If that varies over space and time, IMHO it would be reasonable to expect the cosmological constant to vary over space and time too. 
