It is a known fact that frequent hibernation could slow down a Windows OS due to memory dumps and I have also experienced it myself. Will this be the same in Ubuntu?
Just from personal experience: no. But it should also not do this on Windows. Hibernation by itself should not have any impact on a system since hibernation is only a concern during boots and shutdowns. It basically dumps your machine state to disk during shutdown and loads it back into memory on boot.
A couple of things though: hibernation often does not work out of the box and that is why it is disabled on Ubuntu by default (Suspend works better and was chosen as the default option). It really depends on your hardware, and if it is not compatible it is impossible to get working.
But when it does work it does the job and it will not slow down your system. A hibernation file needs space on a system so a system that is near full does get a performance hit but that is not really a problem with hibernation. A system that needs to swap to get things done will perform less than optimal. That is the same for Linux and Windows but probably more so for a fragmented disk and in the past Linux has had less issues with fragmentation (not sure how Windows is doing in that regard nowadays).