An addition to earlier answers is justified: Bryson's terminology is a little quirky, as well as his proscription being draconian.
We can perhaps understand why he calls the use of hopefully as a pragmatic marker (or, traditionally, sentence adverb) an 'absolute sense' [usage] if we read the following, from http://www.dl.ket.org/latinlit/grammar/ablativeabsolute.htm :
The Ablative Absolute
An Ablative Absolute phrase is used when a thought, condition or
action is grammatically separate but modifies the meaning of the rest
of the sentence. The ablative absolute is sometimes called an
adverbial phrase because it modifies the whole sentence as an adverb
modifies the action of a verb. We use absolute adverbial phrases in
English too: 'They had a pleasant trip, all things considered.'
Here, all things considered is a pragmatic marker, a framing comment by the speaker. The single word thankfully could occupy the same slot. The reference calls the phrase an 'absolute ablative phrase'; Bryson uses the term 'absolute' for the single-word replacement also.
Indeed, in Happy with his burger, Bob didn't notice the man fall off the pier, the usage of the adjective happy is labelled an absolute usage.
Care needs to be taken not to confuse this usage, which applies to a particular way adjectives / adverbs participate in syntactic structures, with the term 'absolute' contrasting with 'comparable':
absolute adjective: An adjective, such as "supreme" or "infinite," with a meaning that is generally not capable of being intensified or
( http://grammar.about.com/od/ab/g/absadjterm.htm )