The maximum is
48 checks.
PGN:
[FEN "8/8/8/8/7k/8/PPPPPPPP/4K3 w KQkq -"] 1. g3+ Kg4 2. h3+ Kf5 3. g4+ Kg5 4. h4+ Kf4 5. e3+ Ke4 6. f3+ Kd5 7. e4+ Ke5 8. f4+ Kd4 9. c3+ Kc4 10. d3+ Kb5 11. c4+ Kc5 12. d4+ Kb4 13. a3+ Ka4 14. b3+ Ka5 15. b4+ Kb6 16. c5+ Kb5 17. a4+ Ka6 18. b5+ Kb7 19. c6+ Kb6 20. a5+ Kc7 21. b6+ Kd6 22. e5+ Ke6 23. d5+ Ke7 24. d6+ Kf7 25. e6+ Kf6 26. g5+ Kg6 27. h5+ Kg7 28. h6+ Kg6 29. f5+ Kh7 30. g6+ Kg8 31. h7+ Kg7 32. h8=B+ Kxh8 33. g7+ Kh7 34. g8=B+ Kg7 35. f6+ Kxg8 36. f7+ Kg7 37. f8=B+ Kxf8 38. e7+ Kf7 39. e8=B+ Kxe8 40. d7+ Ke7 41. d8=B+ Kxd8 42. c7+ Kd7 43. c8=B+ Kxc8 44. b7+ Kc7 45. b8=B+ Kb7 46. a6+ Kxb8 47. a7+ Kb7 48. a8=Q+ Kxa8
Lichess: https://lichess.org/RP8jbdgO
Proof of optimality:
48 is the maximum number of moves allowed by the rules.
Basic idea:
Pushing the pawns one rank at a time at the start is straightforward. To set up the bishop promotion chain at the end, it is helpful to keep the f-pawn behind the g- and h-pawns to avoid interfering.