Well, since photosynthesis is endothermic and the energy is stored inside the plant.
The light that is being used for photosynthesis won't turn into heat.
But photosynth is at most 13% efficient
https://www.researchgate.net/publication/229939005_The_maximum_efficiency_of_photosynthesis
The difference is really small though.
Even if you have a LED system that's literally 100% efficient, and a HPS that's 30% effficient. Both 1000 watt, it's a difference of less than 100 watt.
In reality the difference would be much closer to 30-40 watt.
So you're mostly right, 1000 watt LED box is going to be just about as hot as a 1000 watt HPS box.