Learn German: Der Sommer war dieses Jahr besonders heiß. - The summer was especially hot this year.

Both are actually correct! dieses Jahr uses the accusative case to show a definite time without needing a preposition. Using in diesem Jahr (dative with the preposition in) means exactly the same thing, but native speakers often prefer the shorter dieses Jahr in everyday communication.

The current lesson is about focus particles like sogar or ausgerechnet. Is besonders a focus particle here?

Not quite. Here, besonders acts as an intensifier (a grading particle) rather than a focus particle. Focus particles can often move around the sentence to shift the spotlight to different words (e.g., Sogar der Sommer... vs. ...dieses Jahr sogar heiß). However, because besonders is directly intensifying the adjective heiß, it is locked in place and must sit right in front of the word it modifies.

Could I emphasize the time by starting the sentence with Dieses Jahr? How would that change the word order?

Yes, you can absolutely start with the time expression! If you do, remember the strict "verb-second" rule. The verb war must stay in the second structural position, which pushes the subject to the third position: Dieses Jahr war der Sommer besonders heiß.

Since this describes a past event, why use the Präteritum (war) instead of the Perfekt (ist ... gewesen)?