Empathy can be detected in people whose brains are at rest. Researchers have found that it is possible to assess a person's ability to feel empathy by studying their brain activity while they are resting rather than while they are engaged in specific tasks. Source 4f.
It'd make sense; if I put it logically - doing tasks or activities that pose a certain 'stress' upon the mind causes pressure, making it more selfish and self-centered in terms of having demands to fix it's condition. When that demand is gone, it'd want others to feel what it feels. That being said, if the brain is under stress - perhaps it'd also want others to feel that stress as well?
Empathy may be in the eye of the beholder. Do we always want people to show empathy? Not so, said researchers. A recently published article suggests that although empathy is often portrayed as a virtue, people who express empathy are not necessarily viewed favorably. Source 2b.