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Are there any special rules for invisibility when underwater?
For example, Pathfinder (a different game) has a rule that makes it easier to detect invisible creatures underwater due to the way they displace water (I haven't quoted the text in order to avoid confusion).
Are there any similar rules in D&D 5e?
The Basic rules impose limited restraints on a creature that's underwater: Movement speed is halved for creatures without a swim speed, most melee weapon attacks also have disadvantage for those creatures, and ranged weapon attacks are limited to the 'normal range' of the weapon (i.e. the first number of the two given for weapon range). (Basic rules, p.8)
In fact, the only other rules I know of with any bearing upon hiding underwater are on page 117 of the DMG, under the heading Underwater Visibility. These rules explain that creatures who are underwater will only be able to perceive enemies who are less than a certain distance away, depending on visibility conditions.
5e is light on specific rules for particular cases, and prefers to equip DMs to make rulings in response to the actions of their players. One of the best tools provided for this is the ability to impose disadvantage when 'circumstances hinder success in some way' (DMG p.239).
If a character is attempting to use their invisibility to hide whilst underwater, you could impose disadvantage of their stealth check to reflect their difficulty in so doing*. If spotted, they would still be unseen, but they would no longer be hidden - the main difference is that an enemy would know enough about their approximate location to target them with a disadvantaged attack.
* I'm assuming here that it would be more difficult to remain undetected underwater, since that's the assumption in the question. I have no idea if it's true or not.
Why should there be such a rule? In reality, an object is invisible if it's transparent and has the same index of refraction as the medium it's in. For example, there are polycarbonate plastics that have the same index of refraction as water; they are effectively invisible when placed in water.
If invisibility works by magically making the character transparent and altering their index of refraction, then since the human body is an ugly bag of mostly water, you'd expect that invisibility would work better underwater.
On the other hand, if invisibility works by magically replacing the physical substance of the character with the medium they're surrounded by, then invisibility would still work just as well underwater. Strange things would happen if it were cast while the character is in air and they then moved into water... or the opposite!
Invisibility grants the "invisible" condition, but what exactly does "invisible" mean in 5e? (Emphasis mine)
An invisible creature is impossible to see without the aid of magic or a special sense. For the purpose of hiding, the creature is heavily obscured. The creature's location can be detected by any noise it makes or any tracks it leaves.
Given the creature's noise and possible "tracks" within the water, a viewer could reasonably locate an invisible creature in water.
Otherwise, there are no specifications I am aware of that define DCs or AC bonuses etc regarding this scenario underwater. Only that it is heavily obscured for the purposes of hiding in general.
