I'm not aware of any negative physiological effects of maintaining a steady blood pressure, PROVIDED the pressure is around the regular average a human has today. So, for example, if a normal human has an arterial pressure between 120 mmHg and 80mmHg, their average pressure is 100mmHg. So you couldn't run your synthetic heart at 120mmHg all the time, as that would start to induce the negative changes we associate with hypertension. The higher the continuous pressure, the more negative the effects (typically increased risk of stroke, decreasing kidney function, etc). Of course these devices have lots of other complications, but we'll ignore those (the need for anti-coagulation, etc).
But you have to be able to vary the pressure to some extent because otherwise in a fight or flight response (gotta run!!!) you won't be able to increase blood flow to deliver more oxygen and remove lactic acid. Plus it helps to be able to vary blood pressure when going from laying down to standing up (problems doing this is called orthostatic hypotension). So a continuous pressure heart, inherently, isn't a problem, but one that is LOCKED TO A SPECIFIC PRESSURE is.
Humans vary their blood pressure through adjusting heart rate AND changing the muscular tension on arterial side blood vessels. Venous blood vessels have little to no muscular walls so they can only dilate to compensate for increased pressure or respond to the muscular action of the tissues they are embedded in. If you remove the natural heart, the replacement will have to either be manually controlled, somehow interpret the brains call for increased heart rate by increasing the flow rate and elevating blood pressure (or sense dropping oxygen levels, rising lactic acid, or whatever), or the person would be forced to regulate their activity very carefully lest they deplete oxygen and cramp up due to lactic acid build up. Here is a good technical article discussing various total heart and left ventricular assist devices and exercise tolerance. To my knowledge, all artificial heart type devices meant for prolonged use (i.e. getting out of bed and walking around) replicate or assist a beating heart. ECMO and the like are for sedated/bed-ridden patients, so exercise tolerance is not an issue.