When interest rates rise, we all know that Math shows that the value of a bond will drop, but that's not my question.
Intuition says that there's an indirect risk, in that rising rates might hurt their business, thus increasing the likelihood of bankruptcy.
Any other factors which increase the risk of defaults on existing bonds?
(The purpose of the question is that I'm contemplating what happens -- besides the aforementioned "math" drop in value -- to existing bonds.)
When interest rates rise, existing bonds are not affected from the perspective of a bond issuer (assuming fixed interest bonds). However, when the bond matures the issuer will be forced to repay the face value to the bond holder or they would default. Often this is done by refinancing the existing bond at the current rate.
So when interest rates rise, the bond issuer will be forced to pay a higher interest to maintain the same level of debt. This is slowly increasing the weight of debt on the balance sheet until it can be too much and earnings are insufficient to keep up with interest payments. But this will take a longer time as the weight of debt increases slowly
One common model of measuring default risk is the Merton Model, which does have interest rate as a component. However, that rate is taken as a constant, and is primarily used as a discount factor more than a direct influence on the company's ability to pay debt. There may be other models that treat interest rate as an additional stochastic (randomly changing) variable.
