Cost of adjusting the thermostat
I had my own "green" start up company that I sold off 10 years ago- and I use the same guys to install most new systems I put in.  Really for most cases all the data will come back to is the average temperature that a home is kept (given that insulation and other thermal constraints are constant).  

Really the most effective thing to do is to always have your heat pump on at a constant that would give you the lowest temperature that you could stand.  Most of this comes from the ducts/pipes being used are always heated to level.

There are numerous numerous studies that will show you that if you turn off your heat pump for a large period of time of the day or turn it down dramatically you can save a lot of money.  The energy.gov study shows that turning off/down your thermostat for 8 hours a day will save you between 5-20% of your heating bill.

You might say - well that shows you that it cost more to heat up your house because the heat pump is off for 33% of the day but you only save 10%.  Now remember the point I made earlier, the cost is really based on the average temperature of your house - not when/how long your heat source is on.  So if it is 20F outside and you normally have your thermostat set at 70F but turn it off 8 hours a day you may come back to a house around 59F.  Now your average temperature is 66.33% and you can expect savings of 3-5% per degree around ambient temperature so about 10-20% savings which is directly in line with energy.gov stats.

Now the big GOTCHA that I didn't see data on is the ability for the heat pump to source heat from the outside.  (Let's take your heat pump being old out of the equation.  Most new heat pumps are anywhere between 150-300% efficient.  Older ones are all over the place so this may or may not pertain.)  If it is below 20 or just too cold and you kick your thermostat off for 8 hours a day and your house gets to 59F when you back... now let's say your heat pump then has a hard time getting the house back up to the desired temperature - and yes you would kick it off when you are at work and when you get home with cold house the temp outside is dropping too.   

So the kicker is with a heat pump is you never want to turn it off to the point where you ever need to use the emergency heat.  The emergency heat is probably electric and might be close to 100% efficient for new models, but no where near the normal range of efficiency.  So putting yourself in a situation where the temp fell so much that it kicks on the emergency heat on a heat pump is the very last thing you want to do.

But even in this case it does not mean you don't turn down your thermostat, it just means you only turn it down to a certain level based on the outside temperatures.  (there is a lot more I could have probably added to this but it is already starting to get long)

Physics discussion that is a little more involved on the science side than my answer.

Really good case study by Institute for Research in Construction

There is also a lot of discussion that could be included in maybe a different question that is more specific around:


dealing with different types of heat sources (heat pump vs central air - forced vs rad)
the thermal properties of your house
the way your house deals with heat loss at variable levels of heat differential
the inefficiencies of your heating solution (ie is there anything that makes your heat source inefficient and how to limit that)
the humidity effects and other climate effects