For the record, due to the many instances of this problem on this forum, without any seemingly finding a solution, here is my sum up:

I got the Application Started entry in my Event log every other minute. I sent my web-server provider the following suggestions from this forum:

"You'll probably find that it is in an application pool that is set to recycle on certain memory usage limits. Tell your host that it is restarting every couple of minutes and ask if they could please check and adjust the app pool recycle triggers, or move your domain to a pool that can handle it." 
"See if your hosting provider can give you any information as to what the ASP.NET worker process recycle settings are for your server.  Maybe they have something that is causing it to shutdown a lot." 
"But especially low budget hosting companies tend to put too many sites on one server and then IIS recycles (stop-start) applications if they use too much memory or if they have been inactive for a certain period." 
They increased my memory-recycling max from 40 to 60 MB and now the site is instantly OK, without the many Application started log-entries.
They also informed me that the Recycle worker process is 1740 minutes, and the server-timeout period is 20 minutes.
My DNN version is 5.6.2

Hope this can help others. Cheers! Best regards, OJ, Stavanger, Norway

After 24 hours problems seem now to be over for my site's part. No hyperactive Application started errors. Just to confirm that the solution above seems to work. 

The considerations above regarding memory recycling max and the application pool should maybe be taken into dnn standard documentation as possible issues to be dealt with for those of us having the production site in a shared hosting environment, even for smaller sites with moderate traffic as my own (delfinen.no). 

(I might add that I also have two services hitting my keepalive.aspx file at 15 minutes intervals, just to keep my site from dying. Both were active during the problems encountered though, so these did definitely not contribute to solve the issues I had until yesterday)

Hope you manage to solve your issues Marius. Cheers!