timed automata - advance only in certain states
So, after looking deeper into the subject I found the solution. Posting in case somebody else needs an answer and is unfamiliar with the subject.

What I was looking for is called a Stopwatch Automaton (SWA) and allows the time to stop or advance (by specifying the derivative of a clock c' = {0, 1}).
Reference: The Impressive Power of Stopwatches. Franck Cassez, Kim Larsen
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.123.669&amp;rep=rep1&amp;type=pdf

Further I discovered that Hourglass automata also allow time to stop, but additionally also to reverse: c' = {-1, 0, 1}.
They however require the specification of a maximum clock value after which "all the sand passed through the hourglass" and the clock does not advance further.
Reference: Hourglass Automata.
Yuki Osada, Tim French, Mark Reynolds, Harry Smallbone
https://arxiv.org/abs/1408.5965