Quick answer:
Not all that cold.
Most the mountains in Southeast Asia are close to the equator, and aren’t all that high, at least when compared with the sort of peaks in the Himalayas, Andes, or Rockies. So you can leave your crampons at home.
You do still have to think carefully about your clothing, particularly to manage the humidity and heat of the lower levels, and some chilliness toward the top.
A scenario for some perspective
To give you a very rough idea, let’s take the scenario that its a day with a high of 30 degrees celsius at sea level where you’re about to climb. If so, here are our best guesses of temperatures at your summit depending on it’s height:
Hills: Between 500 and 1,500 m
Estimated daily high at summit: Between 20 and 25 degrees celsius
Examples: Mount Ophir, Mount Lambak, Mount Santubong, Mount Merapi
What to add to your pack: Maybe an extra shirt, in case there’s a chilly breeze. Otherwise your raincoat should add whatever extra warmth you need. Maybe a sweater if you’re camping out overnight.
Small mountains: Between 1,500 and 2,500 m
Estimated daily high at summit: Between 15 and 20 degrees celsius
Examples: Mount Bromo, Pinatubo, Mount Bintan, Mount Tahan, Doi Chiang Dao, Mayon Volcano
What to add to your pack: Maybe a light woolen sweater, or thermal undergarment for your top half. Thermal leggings and a beanie if you’re staying out overnight in a light sleeping bag.
Medium Mountains: Between 2,500 and 3,500 m
Estimated daily high at summit: Between 10 and 15 degrees celsius
Examples: Fan Si Pan, Mount Apo, Mount Agung
What to add to your pack: Thermal underwear, a beanie, and a woolen outer layer or fleece. It will be quite chilly overnight, meaning either a good fleece or windbreaker, some gloves, dry woolie socks, and a good sleeping bag.
Higher Mountains: Higher than 3,500
Estimated daily high at summit: Between 5 and 10 degrees celsius
Examples: Mount Kinabalu, Mount Semeru, Mount Rinjani, Mount Kerinci
What to add to your pack: Thermal underwear, a beanie, and a fleece outer layer, a good windbreaker, and/or proper fleece-lined cold-weather jacket. There are options to camp out overnight at some of these mountains, so be prepared with another woolen middle layer, gloves, dry woolie socks, a good sleeping bag, a tent, and someone warm to snuggle.
Puncak Jaya (Carstensz Pyramid): 5,262 m
Estimated daily high at summit: Between 0 and 5 degrees celsius
What to add to your pack: Puncak Jaya is an alpine climb requiring a lot of gear, advanced mountaineering skills and equipment, and a number of days camping out in its lower reaches before attempting a summit. Best check in with our page devoted to it.
Note: Technically, Hkakabo Razi in Myanmar is the highest mountain in Southeast Asia at an estimated 5,581, but with there being so few climbers (only two who have survived) and with it being very much part of the Himalayas, we don’t cover it.