I should have clarified - you can have multiple appliances if they are exactly the same, such as heat tape on a rack. A heat lamp and UTH should have their own controllers.
Also I have that t-stat, it's an on/off switch type, not a proportional where it adjusts the amount of power. So while fine for a UTH, the OP will see the lights turn off and on all the time as the t-stat works, which would be annoying (to me anyway).