The hawthorn wood, according to wikipedia, is stated as being a possible source for some nasty (and some not so nasty but still worrying) side-effects: "Overdose can cause cardiac arrhythmia and dangerously lower blood pressure. Milder side effects include nausea and sedation". Obviously this is with regards to humans, but it is safe to assume the effects won't be much better for your degus. If your degus are prone to chewing everything they can get hold of, it is probably best to steer clear of this wood just to err on the side of caution.
Chloe @ Degutopia states: "Safe wood for degus includes eating Apple, Hazelnut, Hawthorn and Kiln-dried Pine" and includes the footnote "Safe wood must be Organic, untreated and contain no pesticides, chemical treatments or paint"