I've lived in London for uni for the last 2 years and I'd generally say that most places I've been, excluding those right in the very centre, have actually been cheaper than most places in Southampton.
For anyone who likes London Pride- the Coal Hole on the strand seems to have the nicest pint of Pride I've ever had- not sure why. Oh and it's only £2.50 a pint.
Also if you want just like a 'snacky' lunch, check out Leon's on the strand, their salads are the topps.