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Above all, thanks for responses!

a) By the time we get home this pizza will be cold.
b) By the time we get home this pizza will have been cold.

We were taught that if in the "by the time" clause, we use the simple present,
in the other clause we should use the future perfect. Then, are (a) and (b)
grammatically correct and equal in meaning?

If you have been taught that as an absolute rule, then you have been misinformed!

A future simple form in the main clause is perfectly possible in a sentence beginning 'By the time...', and is, in this case, the only sensible choice. An alternative formulation employing the future perfect, however, would be '...the pizza will have got(ten) cold'.