I found this easier than usual. Solving time: 14 mins.
Notes on some clues below – happy to explain others if asked.
* = anagram
ACROSS
11 TOAST Double definition
13 APPLE T From computers (Java)
27 OUTDO Hidden in withOUT DOubt. Definition = best, the outsiders are the outside letters – with ubt.
DOWN
4 STAR(ving) Definition = somebody (famous)
14 THE RAINBOW (Novel by DH Lawrence) (bra white no)*
17 M (A CAR) OON Moon “made of green cheese”.