My old notes indicate that Åsnes was a parish within Hof, Hedmark at some time.
The LDS Family History Library online catalog lists that there is a bygdebok for the area which is entitled "Åsnesboka" (two volumes? which they have copies of in Salt Lake City). A quick check online also indicates that there are other libraries which hold those volumes and which are willing to loan through Inter-Library Loan program.
You MIGHT also have a chance of finding at least part of those books online through GoogleBooks.
You should also consult the parish records for your family. The detail available in the extensive parish church records is a marvelous gift from Norway. The Digitalarkivet web site has been adding scanned images of the original Norwegian parish church records for anyone with Internet access since November 2005. To be able to use the parish records you must first know WHERE in Norway you want to search for this documentation.
The Digitalarkivet web site - (http://digitalarkivet.uib.no/cgi-win/WebFront.exe?slag=vis&a...
An English option is available by clicking on that word from either the left hand column or the blue link bar along the bottom of the homepage.
The scanned images are available from the homepage link "Skanna kyrkjebøker" [Norwegian version]/"Digitised parish records" [English version] which is listed along the left hand column and from the blue banner of links along the top section of the homepage.
When you click on that link another page will present a short list of choices - choose "Read the digitized parish registers"[English version]/"Lesa skanna kykrebøker" [Norwegian version].
After you've clicked on that link and a new main page has presented on the screen be sure and read the instructions that are available from the Digitalarkivet for navigating the scanned records. The instructions are available in Bokmål (official Norwegian), Nyorsk (Norwegian), Davvisámegiella (Saami), and English.
Recommended basic reading are the "Startsiden" [Norwegian version]/"Main page" [English version], "Brukerveiledning" [Norwegian version]/"User's guide" [English version] and "Om tjenesten" [Norwegian version]/"About this service" [English version].
Good information about translating the formats of Norwegian parish church records during various time periods, many of the basic terms used and understanding how to use the information should be studied at this web site -