Home still warming even though thermostat set to cool

We seem to be having an odd problem with our heat pump/HVAC system. As temperatures here have recently (finally) climbed into the mid to upper 60s, our house heated itself up to about 78 degrees (per the thermostat reading). The thermostat was set on "heat" and set to 72. As the 78 degrees in the house was pretty warm last night, I went to set the thermostat to "cool". I noticed the thermostat was flashing to "replace batteries". I did so. I then set the thermostat to "cool" and to 73 degrees. This morning the temperature still read 78.
This evening I'm fighting the same battle. The termperature inside has actually gone UP to 79. Thermostat still set to cool. I checked to verify that the heat pump outside has kicked on and it has.
The heat pump was replaced in 2001 and the inside system in 1999. Could this be a switch sticking and not cliking over from "heat" to "cool"? Does the system take time to move from heat to cool? I don't remember this from past years, but I also don't rememeber big early March temp swings from 20 to 70 in just a few days (though I'm sure they happened). Could this be a weird result of the thermostat batteris frittering out before we replaced them? Could this be a freon issue? I'd like to get some level of education on this before inviting the HVAC guys out to the house to explain how this is going to cost me some hundreds of dollars.
After I originally posted this in alt.hvac last night, I noticed something else that happened overnight. I turned off the system alotgether since the outside temps were supposed to be right around 50 degrees overnight. The house is still 78 degrees inside even this morning. I have a regular store bought thermometer sitting on top of the thermostat just to doublecheck. I know there's probably some physics lesson out there that could help explain, but I'm curious as to why the house is hanging on at 78 with no mechanical intervention even though it's never gotten that warm outside. Maybe the heat pump wasn't warming the house, but it just wasn't cooling it either.....
Thanks very much for any help.
jeff
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Solar heat!
Hound Dog
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Jeff and Beth wrote:

cooking, the heat from the attic filtering down into the rooms...and any appliance thats plugged in including the computer/tv etc... most of the time in the south we dont need any heat at all...at night the temp. goes down to 55 or so and in the day around noon its about 75 to 80 degrees the last couple of days.. i gotta turn the a/c on to about 75 to just get the a/c to come on... it will cool the house down, but at 78 or 79 it does not come on and the house feels warm.....if i read your post correctly its probably just the variable of what hot and cold to you??? and we always have to ask who put the themostat to 80 or 85??? someone always turns it up if they feel cold when coming out of th bath after taking a shower... and no one ever claims that they did that.. well i know my thermostat does not chang itself.... with a house full of kids it must be the ghost or something that is doing it????
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Turns out heat pump fan was kicking on but compressor was not. Technicians got compressor "bump" started but said it was "noisy and inefficient" and producing very little pressure from the pistons. Said they could replace it (with the physical part itself free since it was under warranty) for $880 plus their $85 fee for the diagnostic visit. Since the weather is nice and I don't REALLY need a/c right now I shopped around for better prices. Eventually found a highly rated place that would do it for $575 (plus they could do it in about a 16 hour turnaround). I asked the the first place to give me a better price but they dug out their files and found that we'd had a snit about the original heat pump price three years ago (same company supplied it; it was a York). They said based on our issues from 3 years ago they were going to stick with their current price (by far the highest of any of my quotes).
Anyway, problem diagnosed and will be solved tomorrow. I hate to spend the money, but I figure it's probably cheaper to have had it happen now rather than in the middle of a heat wave when I'd have likely been stuck paying an even higher price.
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