what does common contract law say on how much Bob actually owes to
Contract law is premised on the principle of the parties knowingly and willfully committing to the clauses at issue. Since the contract is unlikely to provide any guidance on "pricing methodology", the outcome would depend on the extent to which a reasonable person may say each party knowingly and willfully accepted what the final amount would be.
By default, Rob as a provider is expected or presumed to have a better notion of expenses than Bob. From the start of the work and up until just prior to invoicing, Rob (unlike Bob) had more up-to-date information on the costs that ensued during the process.
Therefore, Rob's decision(s) to incur expenses without first seeking Bob's concurrence makes Rob responsible for those costs (or quite a major portion thereof), since it cannot be said that Bob knowingly agreed to an amount which significantly exceeds the estimate. Ruling otherwise would render the term "estimate" meaningless.
In other words, Rob is on the wrong for presuming that
informing Bob of the extra expenses is unnecessary — he will accept
what they factually are.
Had Rob consulted with Bob during the process about the unforeseen expenses, Bob could (1) have been able to direct Rob to cheaper substitutes that would allow Rob to complete the job, or (2) decide to cut losses and pay Rob the amount hitherto earned.