LIES
It does though. The bather weighs more than the combined air and water they’re displacing. That extra weight causes more deflection of the bottom of the tub which increases the volume of the tub (by maybe a mL?).
Admittedly the most pedantic thing I’ve written on this forum but going the the ‘going soap’ theme…
Don’t dismiss that extra mL! We’re dealing with marginal gains here.
Depends what the tub is made of, is it Hooke’s law? 
In general, yeah. Though it depends on the person and whether they’ve exhaled or not. Fat tissue is less dense than water as is the air in the lungs. Most everything else is neutral or more dense. When I was at peak fitness I could stand on the bottom of a pool (after exhaling) without having to scull to stay there. Also I was envisioning a normally filled bath tub; you don’t have to displace much air for the net mass to be greater.
As far as the material, everything bends, everything deflects (or breaks). Even if it’s less than a microscopic amount.
Let’s give TR some credit. On other forums a thread with this title would have been immediately deleted and the OP banned. (Not a suggestion btw, merely a reflection from someone who hangs out in other far more rabid forums.)
This gets to be quite interesting when you are trying to machine and measure things to with a a few millionths of an inch. Even touching a metal bar with your fingers will heat it enough to distort it. There are a variety of interesting videos on this on Youtube, etc.
Your cast iron bathtub will hold more hot water than cold water 
I wasn’t even considering thermal expansion. Now this ‘going soap’ tangent has a whole new lane!
It’s not just the bottom that will be affected, though - the sides are going to deform under the tension too. If the tub has a relatively round cross-section, this could cause a narrowing of the shape (into something more like a V) that results in a reduction of the volume.
Would the v-shaped tub be more of less aero than the rounded shape?
Well I can see how having hot water might cause an expansion of the tub so that itdeforms, but surely if I’m going uphill then adding water is just going to slow me down, even it’s more aero?
I know it’s important to stay hydrated but…
You need to stay hydrated and if you’re going uphill than hot water is important due to the decrease in density. Those milligrams are important.