Carbonfiberboy 

Cadence is very likely what it is. In erg mode, lower cadence should produce a lower HR and vice versa. So put on a cadence sensor. If you don't have a head unit either on your bike or integral with the trainer which will work with a cadence sensor, then there are is a plethora of inexpensive bike computers, both wired and wireless, to choose from.

My HR isn't perfectly stable, not at all. It'll go up and down maybe 3 beats, probably depending on whether I'm in a meditative state or thinking about something, but when I'm in good condition the average will stay steady as long as I don't go over 75% FTP or 70%-75% MHR or ride longer than my current conditioning allows me to hold it steady. It takes me 30' to warm up to a steady HR, so I only measure drift from there on out. When one has drift, it will increase the longer one rides.

I started out using a power where I didn't have any drift in an hour's ride, by experimentation. Then I gradually increased the steady power in the last 45' and then trained enough days to establish enough fitness to get steady HR again at that power. I continued this power increasing and stabilization workout until I had increased power to the above figures. Now, I'm trying to gradually increase the length of these workouts at that same power. My research says that above those effort levels, HR is always going to gradually increase due to lactate buildup.

Erg mode greatly simplifies the exercise but is more boring than having to hold power steady by oneself.

All this said, you can see by the forgoing that this isn't everyone's cup o' tea. Which is a little odd, considering that every training plan says to develop a deep base in early winter. So this year, I'm doing that.