Wouldn't active cloaking devices violate the law of thermodynamics - or be very hot - in Star Trek?

Question

It's been established that cloaking devices such as those used by the Romulans or in the USS Defiant require a great amount of energy to sustain.

But wouldn't the energy have to go somewhere or end up violating at least several laws of thermodynamics. Either the temperature within the cloaked area would have to increase or they would have to dump the energy somewhere?

Note 1: Getting rid of heat is a critical problem on many of our current (21st century) spaceships (ie. satellites). Space isn't cold, just well insulated.

Note 2: If the federation used passive cloaking such as the current class of metamaterials that bend EM radition around themselves, that'd be one thing, but they'd still have to deal with the considerable energy created by the warp core running -- and all those human bodies, life support systems, etc.

Answer 1

The Klingon Bird of Prey Owner's Workshop Manual, which was coauthored by Rick Sternbach (Sternbach and Michael Okuda were the technical consultants on Star Trek: The Next Generation and subsequent shows, and they wrote the Next Generation Technical Manual which is often treated as canonical), describes the Klingon cloak on p. 46:

The cloaking field acts to mask the presence of the Bird-of-Prey through quantum teleportation, by actively transporting matter and energy from outside the envelope to the other side almost entirely unchanged and detectable only with the fastest and most sensitive instruments. Matter and energy produced inside the envelope, from sources like the ship's impulse engines, are temporarily stored within the field. An extremely small fraction of the total energy flux from local space, approximately 0.003 per cent, penetrates the cloak to allow for sub-light navigation.

An EM field produced by the emitter on the ship's exterior performs the actual cloaking by way of a complex spatial phasing of most incoming radiation. This phasing involves the short-range, almost instantaneous, quantum teleportation of radiation and particles through the cloak envelope as well as the ship itself. While the process is not 100 per cent efficient, it works well in the majority of situations a Bird-of-Prey is likely to encounter.

The bolded sentence explains what happens to heat generated from within--it is somehow stored within the field of the cloak itself, which must act as a heat sink while the ship remains cloaked. I would guess the heat can be dumped into space when the ship decloaks.