Does deep, slow breathing increase mental performance? Emotional Intelligence 2.0 said that breathing deeply and slowly increases mental performance, but I couldn't find any studies that really study the claim, and the book is a self-help book, which tend to be extremely unrigorous. So, does it increase mental performance?
I spent some time searching various databases with a variety of different keywords related to breathing techniques and mental outcomes. I only found a couple of studies that have looked at certain Yoga deep breathing techniques and their effects on cognitive tests measuring attention/concentration (Shirley Telles et al. 2007, 2013). Some of these specific techniques (such as selective or alternate nostril breathing) seemed to lead to better performance as compared to control groups (e.g., only breathing awareness). 

I have to admit that I am a bit skeptical because the studies have small samples and placebo effects may have played a role.

In conclusion, there does not seem to be much research that would allow the strong and general claim the self-help book is making.

References

Shirley Telles, P. Raghuraj, Satyapriya Maharana, and H. R. Nagendra (2007) immediate effect of three yoga breathing techniques on performance on a letter-cancellation task. Perceptual and motor skills: volume 104, issue , pp. 1289-1296. 

Shirley Telles, Arti Yadav, Ram Kumar Gupta, and Acharya Balkrishna (2013) Reaction Time Following Yoga Bellows-Type Breathing and Breath Awareness. Perceptual and Motor Skills: Volume 117, Issue , pp. 89-98.
doi: 10.2466/22.25.PMS.117x10z4