Quick update following my discussion with the support team:
They suggested that I rename the folder "Evernote" into "Evernote.old" in the "Application support" folder (I suppose that deleting the folder would have had the same impact). When I opened again the app and logged-in, a new "Evernote" folder was created, which I guess is a bit like forcing a new, clean installation. This did not solve the problem, the CPU was still used like crazy and the computer was still overheating.
They suggested to wait a bit and try to install the app again if things were not improving. I actually did it immediately, by logging-off (+ choosing the option to erase data from my device), deleting the app and re-downloading it, not from the Evernote website this time but directly via the Apple store (does this make a difference ?). The problem seem to be solved now!!!