Correct. The tropics remained tropical. The only effect the ice sheets had on the tropics was to significantly dry them out.
The rainforests/jungles areas of Africa, Southeast Asia and Central/South America were more Savana-like and even grasslands. Existing tropical deserts were larger, or due to changes in monsoon rains patterns, grassland.
The repeated aridity of East Africa led to humanity as the remaining apes shrunk with their forests. The weaker apes with the weird feet and long legs were better adapted to the grass.