What causes reverse leakage current in an LED? I've read it can be due to 'intrinsic semiconductor conduction' and surface conduction. The LED I am looking at is InAs, which is apparently prone to surface conduction. I have tried out what these two quantities are on google but have been unsuccessful. In particular because I have no extensive mathematical knowledge of the subject, and so am looking for a qualitative answer. 
And why does reverse leakage current increase with higher temperatures? 
 A: The static electric field inherent within all diodes (and therefore LEDs) along with phonon (heat) interactions with the lattice is what causes reverse leakage current. LEDs are generally diodes with specific band-gaps so from here on, I'll just talk about diodes.
Diodes are pn junctions. A pn junction in the simplest sense is a collection of static positive charges on one side and a collection of static negative charges on the other side. In this small region between these static negative charges and static positive charges is an area where if an electron-hole pair is created, the electron will head towards the positive static charges and the hole will head toward the negative static charges. This would create a small current if we connected the terminals of the device together.
There's a couple things that are well known to create electron-hole pairs in a lattice.
1) photons hitting electrons exciting them to higher energy states across the band gap.
2) phonons (waves caused by heat) hitting electrons exciting them to higher energy states across the band gap.  
Photons are generally much more energetic so we utilize this phenomenon called the photoelectric effect in solar cells. Phonons on the other hand tend to be of much lower energy so you only get a minuscule amount of current because most of them simply don't have enough energy to breach the bandgap of the semiconductor. With higher temperature, the band-gap narrows and the energy of the phonons increases which will increase your reverse leakage current.
If you wanted to decrease this current, you could do three things, increase the band gap you're using, cool your device down, or decrease the width of your pn junction. In all cases, you'll reduce the amount of electron-hole pairs being generated within the region which will reduce your current.
A: As Horta answered, there are a few pathways for current leakage in a classical pn junction. Both phonon and photon generated carriers contribute to leakage current.
 Look up the derivation of the ideal diode equation. Forward bias current is due to diffusion, the exponential term. Reverse bias current , the constant term, is due to drift of generated carriers by the built in field of the junction. At equilibrium, they balance.
The key here is the word "ideal". If that equation held everywhere, you could bias to negative infinity with no additional current.  Other processes occur at high reverse field. These include Zener tunneling, defect assisted tunnelling, Avalanche breakdown, and dielectric breakdown. They all contribute at some point, and will cause "leakage" current. These processes aren't included in the ideal diode equation!
