I'll take the plunge into pedantry here
Forever.
Now, I'd qualify this as it would need good men at the tiller and luck in those aspects they cannot control.
Could the form and function of the republic change? Sure. It could become 'federal', with lesser Senates providing representation to Rome. A very slow legislative body that acts underneath the military leadership. It could become a republic in that the senate elects the Emperor, but controls funding, and has oaths sworn to uphold the Senate over the Emperor.
Hell, it is at least republican, but you could have a series of 'Brother Republics united by the Emperor' deal going on. The Roman, Gallic, Hispanic, etc, etc - all self-funded republics in their own right, with a common Emperor, sort of like a united Roman Commonwealth.
But I'm being a pedant on the word 'could' there.
