I tried -sdlvideofps with both -video opengl and -video soft and

Code
./mame64 apple2e -sdlvideofps -video opengl
  0.02s,    1 F, avrg game: 60.09 FPS 16.64 ms/f, avrg video: 115.03 FPS 8.69 ms/f, last video:   inf FPS 4.33 ms/f
  1.02s,   57 F, avrg game: 55.82 FPS 17.91 ms/f, avrg video: 169.71 FPS 5.89 ms/f, last video:   inf FPS 1.17 ms/f
  2.04s,  115 F, avrg game: 56.45 FPS 17.71 ms/f, avrg video: 164.82 FPS 6.07 ms/f, last video:   inf FPS 2.71 ms/f
  3.04s,  171 F, avrg game: 56.27 FPS 17.77 ms/f, avrg video: 165.24 FPS 6.05 ms/f, last video:   inf FPS 1.96 ms/f
  4.05s,  230 F, avrg game: 56.74 FPS 17.62 ms/f, avrg video: 170.18 FPS 5.88 ms/f, last video:   inf FPS 16.02 ms/f
  5.07s,  289 F, avrg game: 57.04 FPS 17.53 ms/f, avrg video: 184.40 FPS 5.42 ms/f, last video:   inf FPS 7.71 ms/f
  6.08s,  349 F, avrg game: 57.40 FPS 17.42 ms/f, avrg video: 196.23 FPS 5.10 ms/f, last video:   inf FPS 1.16 ms/f
  7.09s,  408 F, avrg game: 57.53 FPS 17.38 ms/f, avrg video: 193.83 FPS 5.16 ms/f, last video:   inf FPS 15.00 ms/f

./mame64 apple2e -video soft
  0.01s,    1 F, avrg game: 78.69 FPS 12.71 ms/f, avrg video: 75.66 FPS 13.22 ms/f, last video:   inf FPS 6.12 ms/f
  1.02s,   59 F, avrg game: 57.62 FPS 17.36 ms/f, avrg video: 135.10 FPS 7.40 ms/f, last video:   inf FPS 5.98 ms/f
  2.04s,  117 F, avrg game: 57.42 FPS 17.41 ms/f, avrg video: 121.47 FPS 8.23 ms/f, last video:   inf FPS 6.30 ms/f
  3.06s,  175 F, avrg game: 57.12 FPS 17.51 ms/f, avrg video: 120.19 FPS 8.32 ms/f, last video:   inf FPS 19.81 ms/f
  4.07s,  231 F, avrg game: 56.75 FPS 17.62 ms/f, avrg video: 114.54 FPS 8.73 ms/f, last video:   inf FPS 29.81 ms/f
  5.08s,  289 F, avrg game: 56.92 FPS 17.57 ms/f, avrg video: 114.66 FPS 8.72 ms/f, last video:   inf FPS 6.87 ms/f
  6.10s,  348 F, avrg game: 57.02 FPS 17.54 ms/f, avrg video: 112.14 FPS 8.92 ms/f, last video:   inf FPS 19.09 ms/f
  7.12s,  408 F, avrg game: 57.33 FPS 17.44 ms/f, avrg video: 115.75 FPS 8.64 ms/f, last video:   inf FPS 6.46 ms/f


-nowaitvsync doesn't seem to make any difference.

lspci tells me that I'm running a:
01:00.0 VGA compatible controller: Advanced Micro Devices, Inc. [AMD/ATI] RV710 [Radeon HD 4550]

glxinfo | grep AMD
Device: AMD RV710 (DRM 2.49.0 / 4.10.0-33-generic, LLVM 4.0.0) (0x9540)
OpenGL renderer string: Gallium 0.4 on AMD RV710 (DRM 2.49.0 / 4.10.0-33-generic, LLVM 4.0.0)